diff --git a/Lib/test/string_tests.py b/Lib/test/string_tests.py
index 1b5db94..4800d6d 100644
--- a/Lib/test/string_tests.py
+++ b/Lib/test/string_tests.py
@@ -5,7 +5,6 @@
 import unittest, string, sys, struct
 from test import support
 from collections import UserList
-import _testcapi
 
 class Sequence:
     def __init__(self, seq='wxyz'): self.seq = seq
@@ -1185,20 +1184,28 @@
         # Outrageously large width or precision should raise ValueError.
         self.checkraises(ValueError, '%%%df' % (2**64), '__mod__', (3.2))
         self.checkraises(ValueError, '%%.%df' % (2**64), '__mod__', (3.2))
-
         self.checkraises(OverflowError, '%*s', '__mod__',
-                         (_testcapi.PY_SSIZE_T_MAX + 1, ''))
+                         (sys.maxsize + 1, ''))
         self.checkraises(OverflowError, '%.*f', '__mod__',
-                         (_testcapi.INT_MAX + 1, 1. / 7))
-        # Issue 15989
-        self.checkraises(OverflowError, '%*s', '__mod__',
-                         (1 << (_testcapi.PY_SSIZE_T_MAX.bit_length() + 1), ''))
-        self.checkraises(OverflowError, '%.*f', '__mod__',
-                         (_testcapi.UINT_MAX + 1, 1. / 7))
+                         (sys.maxsize + 1, 1. / 7))
 
         class X(object): pass
         self.checkraises(TypeError, 'abc', '__mod__', X())
 
+    @support.cpython_only
+    def test_formatting_c_limits(self):
+        from _testcapi import PY_SSIZE_T_MAX, INT_MAX, UINT_MAX
+        SIZE_MAX = (1 << (PY_SSIZE_T_MAX.bit_length() + 1)) - 1
+        self.checkraises(OverflowError, '%*s', '__mod__',
+                         (PY_SSIZE_T_MAX + 1, ''))
+        self.checkraises(OverflowError, '%.*f', '__mod__',
+                         (INT_MAX + 1, 1. / 7))
+        # Issue 15989
+        self.checkraises(OverflowError, '%*s', '__mod__',
+                         (SIZE_MAX + 1, ''))
+        self.checkraises(OverflowError, '%.*f', '__mod__',
+                         (UINT_MAX + 1, 1. / 7))
+
     def test_floatformatting(self):
         # float formatting
         for prec in range(100):
diff --git a/Lib/test/support/__init__.py b/Lib/test/support/__init__.py
index 233b13a..5c03f54 100644
--- a/Lib/test/support/__init__.py
+++ b/Lib/test/support/__init__.py
@@ -25,7 +25,6 @@
 import logging.handlers
 import struct
 import tempfile
-import _testcapi
 
 try:
     import _thread, threading
@@ -1349,6 +1348,7 @@
 _TPFLAGS_HEAPTYPE = 1<<9
 
 def check_sizeof(test, o, size):
+    import _testcapi
     result = sys.getsizeof(o)
     # add GC header size
     if ((type(o) == type) and (o.__flags__ & _TPFLAGS_HEAPTYPE) or\
diff --git a/Lib/test/test_capi.py b/Lib/test/test_capi.py
index 9013a7b..50c4bba 100644
--- a/Lib/test/test_capi.py
+++ b/Lib/test/test_capi.py
@@ -17,7 +17,8 @@
     import threading
 except ImportError:
     threading = None
-import _testcapi
+# Skip this test if the _testcapi module isn't available.
+_testcapi = support.import_module('_testcapi')
 
 
 def testfunction(self):
diff --git a/Lib/test/test_code.py b/Lib/test/test_code.py
index 3377a7b..21b12a5 100644
--- a/Lib/test/test_code.py
+++ b/Lib/test/test_code.py
@@ -104,7 +104,7 @@
 
 import unittest
 import weakref
-import _testcapi
+from test.support import run_doctest, run_unittest, cpython_only
 
 
 def consts(t):
@@ -126,7 +126,9 @@
 
 class CodeTest(unittest.TestCase):
 
+    @cpython_only
     def test_newempty(self):
+        import _testcapi
         co = _testcapi.code_newempty("filename", "funcname", 15)
         self.assertEqual(co.co_filename, "filename")
         self.assertEqual(co.co_name, "funcname")
@@ -159,7 +161,6 @@
 
 
 def test_main(verbose=None):
-    from test.support import run_doctest, run_unittest
     from test import test_code
     run_doctest(test_code, verbose)
     run_unittest(CodeTest, CodeWeakRefTest)
diff --git a/Lib/test/test_codecs.py b/Lib/test/test_codecs.py
index 76e6dbd..2c6dce7 100644
--- a/Lib/test/test_codecs.py
+++ b/Lib/test/test_codecs.py
@@ -1,4 +1,3 @@
-import _testcapi
 import codecs
 import io
 import locale
@@ -1710,7 +1709,7 @@
 
 class BasicUnicodeTest(unittest.TestCase, MixInCheckStateHandling):
     def test_basics(self):
-        s = "abc123" # all codecs should be able to encode these
+        s = "abc123"  # all codecs should be able to encode these
         for encoding in all_unicode_encodings:
             name = codecs.lookup(encoding).name
             if encoding.endswith("_codec"):
@@ -1722,9 +1721,9 @@
             with support.check_warnings():
                 # unicode-internal has been deprecated
                 (b, size) = codecs.getencoder(encoding)(s)
-                self.assertEqual(size, len(s), "%r != %r (encoding=%r)" % (size, len(s), encoding))
+                self.assertEqual(size, len(s), "encoding=%r" % encoding)
                 (chars, size) = codecs.getdecoder(encoding)(b)
-                self.assertEqual(chars, s, "%r != %r (encoding=%r)" % (chars, s, encoding))
+                self.assertEqual(chars, s, "encoding=%r" % encoding)
 
             if encoding not in broken_unicode_with_streams:
                 # check stream reader/writer
@@ -1742,15 +1741,13 @@
                 for c in encodedresult:
                     q.write(bytes([c]))
                     decodedresult += reader.read()
-                self.assertEqual(decodedresult, s, "%r != %r (encoding=%r)" % (decodedresult, s, encoding))
+                self.assertEqual(decodedresult, s, "encoding=%r" % encoding)
 
             if encoding not in broken_incremental_coders:
-                # check incremental decoder/encoder (fetched via the Python
-                # and C API) and iterencode()/iterdecode()
+                # check incremental decoder/encoder and iterencode()/iterdecode()
                 try:
                     encoder = codecs.getincrementalencoder(encoding)()
-                    cencoder = _testcapi.codec_incrementalencoder(encoding)
-                except LookupError: # no IncrementalEncoder
+                except LookupError:  # no IncrementalEncoder
                     pass
                 else:
                     # check incremental decoder/encoder
@@ -1763,45 +1760,71 @@
                     for c in encodedresult:
                         decodedresult += decoder.decode(bytes([c]))
                     decodedresult += decoder.decode(b"", True)
-                    self.assertEqual(decodedresult, s, "%r != %r (encoding=%r)" % (decodedresult, s, encoding))
-
-                    # check C API
-                    encodedresult = b""
-                    for c in s:
-                        encodedresult += cencoder.encode(c)
-                    encodedresult += cencoder.encode("", True)
-                    cdecoder = _testcapi.codec_incrementaldecoder(encoding)
-                    decodedresult = ""
-                    for c in encodedresult:
-                        decodedresult += cdecoder.decode(bytes([c]))
-                    decodedresult += cdecoder.decode(b"", True)
-                    self.assertEqual(decodedresult, s, "%r != %r (encoding=%r)" % (decodedresult, s, encoding))
+                    self.assertEqual(decodedresult, s,
+                                     "encoding=%r" % encoding)
 
                     # check iterencode()/iterdecode()
-                    result = "".join(codecs.iterdecode(codecs.iterencode(s, encoding), encoding))
-                    self.assertEqual(result, s, "%r != %r (encoding=%r)" % (result, s, encoding))
+                    result = "".join(codecs.iterdecode(
+                            codecs.iterencode(s, encoding), encoding))
+                    self.assertEqual(result, s, "encoding=%r" % encoding)
 
                     # check iterencode()/iterdecode() with empty string
-                    result = "".join(codecs.iterdecode(codecs.iterencode("", encoding), encoding))
+                    result = "".join(codecs.iterdecode(
+                            codecs.iterencode("", encoding), encoding))
                     self.assertEqual(result, "")
 
                 if encoding not in ("idna", "mbcs"):
                     # check incremental decoder/encoder with errors argument
                     try:
                         encoder = codecs.getincrementalencoder(encoding)("ignore")
-                        cencoder = _testcapi.codec_incrementalencoder(encoding, "ignore")
-                    except LookupError: # no IncrementalEncoder
+                    except LookupError:  # no IncrementalEncoder
                         pass
                     else:
                         encodedresult = b"".join(encoder.encode(c) for c in s)
                         decoder = codecs.getincrementaldecoder(encoding)("ignore")
-                        decodedresult = "".join(decoder.decode(bytes([c])) for c in encodedresult)
-                        self.assertEqual(decodedresult, s, "%r != %r (encoding=%r)" % (decodedresult, s, encoding))
+                        decodedresult = "".join(decoder.decode(bytes([c]))
+                                                for c in encodedresult)
+                        self.assertEqual(decodedresult, s,
+                                         "encoding=%r" % encoding)
 
+    @support.cpython_only
+    def test_basics_capi(self):
+        from _testcapi import codec_incrementalencoder, codec_incrementaldecoder
+        s = "abc123"  # all codecs should be able to encode these
+        for encoding in all_unicode_encodings:
+            if encoding not in broken_incremental_coders:
+                # check incremental decoder/encoder (fetched via the C API)
+                try:
+                    cencoder = codec_incrementalencoder(encoding)
+                except LookupError:  # no IncrementalEncoder
+                    pass
+                else:
+                    # check C API
+                    encodedresult = b""
+                    for c in s:
+                        encodedresult += cencoder.encode(c)
+                    encodedresult += cencoder.encode("", True)
+                    cdecoder = codec_incrementaldecoder(encoding)
+                    decodedresult = ""
+                    for c in encodedresult:
+                        decodedresult += cdecoder.decode(bytes([c]))
+                    decodedresult += cdecoder.decode(b"", True)
+                    self.assertEqual(decodedresult, s,
+                                     "encoding=%r" % encoding)
+
+                if encoding not in ("idna", "mbcs"):
+                    # check incremental decoder/encoder with errors argument
+                    try:
+                        cencoder = codec_incrementalencoder(encoding, "ignore")
+                    except LookupError:  # no IncrementalEncoder
+                        pass
+                    else:
                         encodedresult = b"".join(cencoder.encode(c) for c in s)
-                        cdecoder = _testcapi.codec_incrementaldecoder(encoding, "ignore")
-                        decodedresult = "".join(cdecoder.decode(bytes([c])) for c in encodedresult)
-                        self.assertEqual(decodedresult, s, "%r != %r (encoding=%r)" % (decodedresult, s, encoding))
+                        cdecoder = codec_incrementaldecoder(encoding, "ignore")
+                        decodedresult = "".join(cdecoder.decode(bytes([c]))
+                                                for c in encodedresult)
+                        self.assertEqual(decodedresult, s,
+                                         "encoding=%r" % encoding)
 
     def test_seek(self):
         # all codecs should be able to encode these
diff --git a/Lib/test/test_descr.py b/Lib/test/test_descr.py
index 8940ae9..1a891a8 100644
--- a/Lib/test/test_descr.py
+++ b/Lib/test/test_descr.py
@@ -2046,6 +2046,7 @@
             prop2 = property(fset=setter)
             self.assertEqual(prop2.__doc__, None)
 
+    @support.cpython_only
     def test_testcapi_no_segfault(self):
         # this segfaulted in 2.5b2
         try:
diff --git a/Lib/test/test_devpoll.py b/Lib/test/test_devpoll.py
index 819d9dc..9129ac0 100644
--- a/Lib/test/test_devpoll.py
+++ b/Lib/test/test_devpoll.py
@@ -3,8 +3,7 @@
 # Initial tests are copied as is from "test_poll.py"
 
 import os, select, random, unittest, sys
-from test.support import TESTFN, run_unittest
-from _testcapi import USHRT_MAX
+from test.support import TESTFN, run_unittest, cpython_only
 
 try:
     select.devpoll
@@ -94,8 +93,18 @@
         pollster.register(w)
         # Issue #17919
         self.assertRaises(OverflowError, pollster.register, 0, -1)
-        self.assertRaises(OverflowError, pollster.register, 0, USHRT_MAX + 1)
+        self.assertRaises(OverflowError, pollster.register, 0, 1 << 64)
         self.assertRaises(OverflowError, pollster.modify, 1, -1)
+        self.assertRaises(OverflowError, pollster.modify, 1, 1 << 64)
+
+    @cpython_only
+    def test_events_mask_overflow_c_limits(self):
+        from _testcapi import USHRT_MAX
+        pollster = select.devpoll()
+        w, r = os.pipe()
+        pollster.register(w)
+        # Issue #17919
+        self.assertRaises(OverflowError, pollster.register, 0, USHRT_MAX + 1)
         self.assertRaises(OverflowError, pollster.modify, 1, USHRT_MAX + 1)
 
 
diff --git a/Lib/test/test_exceptions.py b/Lib/test/test_exceptions.py
index 43fc7ee..8d11d90 100644
--- a/Lib/test/test_exceptions.py
+++ b/Lib/test/test_exceptions.py
@@ -832,6 +832,7 @@
         self.assertIn("maximum recursion depth exceeded", str(v))
 
 
+    @cpython_only
     def test_MemoryError(self):
         # PyErr_NoMemory always raises the same exception instance.
         # Check that the traceback is not doubled.
@@ -890,6 +891,7 @@
         self.assertEqual(error5.a, 1)
         self.assertEqual(error5.__doc__, "")
 
+    @cpython_only
     def test_memory_error_cleanup(self):
         # Issue #5437: preallocated MemoryError instances should not keep
         # traceback objects alive.
diff --git a/Lib/test/test_fcntl.py b/Lib/test/test_fcntl.py
index b8cda2f..1810c4e 100644
--- a/Lib/test/test_fcntl.py
+++ b/Lib/test/test_fcntl.py
@@ -7,9 +7,9 @@
 import os
 import struct
 import sys
-import _testcapi
 import unittest
-from test.support import verbose, TESTFN, unlink, run_unittest, import_module
+from test.support import (verbose, TESTFN, unlink, run_unittest, import_module,
+                          cpython_only)
 
 # Skip test if no fcntl module.
 fcntl = import_module('fcntl')
@@ -50,6 +50,12 @@
 
 lockdata = get_lockdata()
 
+class BadFile:
+    def __init__(self, fn):
+        self.fn = fn
+    def fileno(self):
+        return self.fn
+
 class TestFcntl(unittest.TestCase):
 
     def setUp(self):
@@ -81,24 +87,27 @@
         self.f.close()
 
     def test_fcntl_bad_file(self):
-        class F:
-            def __init__(self, fn):
-                self.fn = fn
-            def fileno(self):
-                return self.fn
-        self.assertRaises(ValueError, fcntl.fcntl, -1, fcntl.F_SETFL, os.O_NONBLOCK)
-        self.assertRaises(ValueError, fcntl.fcntl, F(-1), fcntl.F_SETFL, os.O_NONBLOCK)
-        self.assertRaises(TypeError, fcntl.fcntl, 'spam', fcntl.F_SETFL, os.O_NONBLOCK)
-        self.assertRaises(TypeError, fcntl.fcntl, F('spam'), fcntl.F_SETFL, os.O_NONBLOCK)
+        with self.assertRaises(ValueError):
+            fcntl.fcntl(-1, fcntl.F_SETFL, os.O_NONBLOCK)
+        with self.assertRaises(ValueError):
+            fcntl.fcntl(BadFile(-1), fcntl.F_SETFL, os.O_NONBLOCK)
+        with self.assertRaises(TypeError):
+            fcntl.fcntl('spam', fcntl.F_SETFL, os.O_NONBLOCK)
+        with self.assertRaises(TypeError):
+            fcntl.fcntl(BadFile('spam'), fcntl.F_SETFL, os.O_NONBLOCK)
+
+    @cpython_only
+    def test_fcntl_bad_file_overflow(self):
+        from _testcapi import INT_MAX, INT_MIN
         # Issue 15989
-        self.assertRaises(OverflowError, fcntl.fcntl, _testcapi.INT_MAX + 1,
-                                                      fcntl.F_SETFL, os.O_NONBLOCK)
-        self.assertRaises(OverflowError, fcntl.fcntl, F(_testcapi.INT_MAX + 1),
-                                                      fcntl.F_SETFL, os.O_NONBLOCK)
-        self.assertRaises(OverflowError, fcntl.fcntl, _testcapi.INT_MIN - 1,
-                                                      fcntl.F_SETFL, os.O_NONBLOCK)
-        self.assertRaises(OverflowError, fcntl.fcntl, F(_testcapi.INT_MIN - 1),
-                                                      fcntl.F_SETFL, os.O_NONBLOCK)
+        with self.assertRaises(OverflowError):
+            fcntl.fcntl(INT_MAX + 1, fcntl.F_SETFL, os.O_NONBLOCK)
+        with self.assertRaises(OverflowError):
+            fcntl.fcntl(BadFile(INT_MAX + 1), fcntl.F_SETFL, os.O_NONBLOCK)
+        with self.assertRaises(OverflowError):
+            fcntl.fcntl(INT_MIN - 1, fcntl.F_SETFL, os.O_NONBLOCK)
+        with self.assertRaises(OverflowError):
+            fcntl.fcntl(BadFile(INT_MIN - 1), fcntl.F_SETFL, os.O_NONBLOCK)
 
     @unittest.skipIf(
         platform.machine().startswith('arm') and platform.system() == 'Linux',
diff --git a/Lib/test/test_fileio.py b/Lib/test/test_fileio.py
index a6853a1..444be91 100644
--- a/Lib/test/test_fileio.py
+++ b/Lib/test/test_fileio.py
@@ -8,9 +8,8 @@
 from array import array
 from weakref import proxy
 from functools import wraps
-import _testcapi
 
-from test.support import TESTFN, check_warnings, run_unittest, make_bad_fd
+from test.support import TESTFN, check_warnings, run_unittest, make_bad_fd, cpython_only
 from collections import UserList
 
 from _io import FileIO as _FileIO
@@ -362,7 +361,11 @@
         if sys.platform == 'win32':
             import msvcrt
             self.assertRaises(IOError, msvcrt.get_osfhandle, make_bad_fd())
+
+    @cpython_only
+    def testInvalidFd_overflow(self):
         # Issue 15989
+        import _testcapi
         self.assertRaises(TypeError, _FileIO, _testcapi.INT_MAX + 1)
         self.assertRaises(TypeError, _FileIO, _testcapi.INT_MIN - 1)
 
diff --git a/Lib/test/test_format.py b/Lib/test/test_format.py
index bd159f5..ea5a731 100644
--- a/Lib/test/test_format.py
+++ b/Lib/test/test_format.py
@@ -313,21 +313,31 @@
     support.run_unittest(FormatTest)
 
     def test_precision(self):
-        INT_MAX = 2147483647
-
         f = 1.2
         self.assertEqual(format(f, ".0f"), "1")
         self.assertEqual(format(f, ".3f"), "1.200")
         with self.assertRaises(ValueError) as cm:
-            format(f, ".%sf" % (INT_MAX + 1))
+            format(f, ".%sf" % (sys.maxsize + 1))
         self.assertEqual(str(cm.exception), "precision too big")
 
         c = complex(f)
-        self.assertEqual(format(f, ".0f"), "1")
-        self.assertEqual(format(f, ".3f"), "1.200")
+        self.assertEqual(format(c, ".0f"), "1+0j")
+        self.assertEqual(format(c, ".3f"), "1.200+0.000j")
+        with self.assertRaises(ValueError) as cm:
+            format(c, ".%sf" % (sys.maxsize + 1))
+        self.assertEqual(str(cm.exception), "precision too big")
+
+    @support.cpython_only
+    def test_precision_c_limits(self):
+        from _testcapi import INT_MAX
+
+        f = 1.2
         with self.assertRaises(ValueError) as cm:
             format(f, ".%sf" % (INT_MAX + 1))
-        self.assertEqual(str(cm.exception), "precision too big")
+
+        c = complex(f)
+        with self.assertRaises(ValueError) as cm:
+            format(c, ".%sf" % (INT_MAX + 1))
 
 
 if __name__ == "__main__":
diff --git a/Lib/test/test_getargs2.py b/Lib/test/test_getargs2.py
index 715b5c6..fe64ee1 100644
--- a/Lib/test/test_getargs2.py
+++ b/Lib/test/test_getargs2.py
@@ -1,5 +1,7 @@
 import unittest
 from test import support
+# Skip this test if the _testcapi module isn't available.
+support.import_module('_testcapi')
 from _testcapi import getargs_keywords, getargs_keyword_only
 
 """
diff --git a/Lib/test/test_io.py b/Lib/test/test_io.py
index e843cac..5c2c79b 100644
--- a/Lib/test/test_io.py
+++ b/Lib/test/test_io.py
@@ -32,7 +32,6 @@
 import unittest
 import warnings
 import weakref
-import _testcapi
 from collections import deque, UserList
 from itertools import cycle, count
 from test import support
@@ -1977,8 +1976,10 @@
             os.environ.clear()
             os.environ.update(old_environ)
 
-    # Issue 15989
+    @support.cpython_only
     def test_device_encoding(self):
+        # Issue 15989
+        import _testcapi
         b = self.BytesIO()
         b.fileno = lambda: _testcapi.INT_MAX + 1
         self.assertRaises(OverflowError, self.TextIOWrapper, b)
diff --git a/Lib/test/test_poll.py b/Lib/test/test_poll.py
index 3c86ef6..a07a719 100644
--- a/Lib/test/test_poll.py
+++ b/Lib/test/test_poll.py
@@ -3,14 +3,13 @@
 import os
 import random
 import select
-from _testcapi import USHRT_MAX, INT_MAX, UINT_MAX
 try:
     import threading
 except ImportError:
     threading = None
 import time
 import unittest
-from test.support import TESTFN, run_unittest, reap_threads
+from test.support import TESTFN, run_unittest, reap_threads, cpython_only
 
 try:
     select.poll
@@ -161,8 +160,18 @@
 
         # Issues #15989, #17919
         self.assertRaises(OverflowError, pollster.register, 0, -1)
-        self.assertRaises(OverflowError, pollster.register, 0, USHRT_MAX + 1)
+        self.assertRaises(OverflowError, pollster.register, 0, 1 << 64)
         self.assertRaises(OverflowError, pollster.modify, 1, -1)
+        self.assertRaises(OverflowError, pollster.modify, 1, 1 << 64)
+
+    @cpython_only
+    def test_poll_c_limits(self):
+        from _testcapi import USHRT_MAX, INT_MAX, UINT_MAX
+        pollster = select.poll()
+        pollster.register(1)
+
+        # Issues #15989, #17919
+        self.assertRaises(OverflowError, pollster.register, 0, USHRT_MAX + 1)
         self.assertRaises(OverflowError, pollster.modify, 1, USHRT_MAX + 1)
         self.assertRaises(OverflowError, pollster.poll, INT_MAX + 1)
         self.assertRaises(OverflowError, pollster.poll, UINT_MAX + 1)
diff --git a/Lib/test/test_posix.py b/Lib/test/test_posix.py
index 60806aa..9408e5c 100644
--- a/Lib/test/test_posix.py
+++ b/Lib/test/test_posix.py
@@ -17,7 +17,6 @@
 import tempfile
 import unittest
 import warnings
-import _testcapi
 
 _DUMMY_SYMLINK = os.path.join(tempfile.gettempdir(),
                               support.TESTFN + '-dummy-symlink')
@@ -615,7 +614,12 @@
         except OSError:
             pass
 
+    @support.cpython_only
+    @unittest.skipUnless(hasattr(os, 'pipe2'), "test needs os.pipe2()")
+    @support.requires_linux_version(2, 6, 27)
+    def test_pipe2_c_limits(self):
         # Issue 15989
+        import _testcapi
         self.assertRaises(OverflowError, os.pipe2, _testcapi.INT_MAX + 1)
         self.assertRaises(OverflowError, os.pipe2, _testcapi.UINT_MAX + 1)
 
diff --git a/Lib/test/test_socket.py b/Lib/test/test_socket.py
index 001712f..67ff1be 100644
--- a/Lib/test/test_socket.py
+++ b/Lib/test/test_socket.py
@@ -7,7 +7,6 @@
 import socket
 import select
 import tempfile
-import _testcapi
 import time
 import traceback
 import queue
@@ -1274,7 +1273,10 @@
             srv.listen(backlog)
             srv.close()
 
+    @support.cpython_only
+    def test_listen_backlog_overflow(self):
         # Issue 15989
+        import _testcapi
         srv = socket.socket(socket.AF_INET, socket.SOCK_STREAM)
         srv.bind((HOST, 0))
         self.assertRaises(OverflowError, srv.listen, _testcapi.INT_MAX + 1)
@@ -1593,6 +1595,14 @@
 
     def _testShutdown(self):
         self.serv_conn.send(MSG)
+        self.serv_conn.shutdown(2)
+
+    testShutdown_overflow = support.cpython_only(testShutdown)
+
+    @support.cpython_only
+    def _testShutdown_overflow(self):
+        import _testcapi
+        self.serv_conn.send(MSG)
         # Issue 15989
         self.assertRaises(OverflowError, self.serv_conn.shutdown,
                           _testcapi.INT_MAX + 1)
@@ -2361,7 +2371,12 @@
     # code with these functions.
 
     # Match the definition in socketmodule.c
-    socklen_t_limit = min(0x7fffffff, _testcapi.INT_MAX)
+    try:
+        import _testcapi
+    except ImportError:
+        socklen_t_limit = 0x7fffffff
+    else:
+        socklen_t_limit = min(0x7fffffff, _testcapi.INT_MAX)
 
     @requireAttrs(socket, "CMSG_LEN")
     def testCMSG_LEN(self):
@@ -3586,14 +3601,23 @@
             pass
         end = time.time()
         self.assertTrue((end - start) < 1.0, "Error setting non-blocking mode.")
-        # Issue 15989
-        if _testcapi.UINT_MAX < _testcapi.ULONG_MAX:
-            self.serv.setblocking(_testcapi.UINT_MAX + 1)
-            self.assertIsNone(self.serv.gettimeout())
 
     def _testSetBlocking(self):
         pass
 
+    @support.cpython_only
+    def testSetBlocking_overflow(self):
+        # Issue 15989
+        import _testcapi
+        if _testcapi.UINT_MAX >= _testcapi.ULONG_MAX:
+            self.skipTest('needs UINT_MAX < ULONG_MAX')
+        self.serv.setblocking(False)
+        self.assertEqual(self.serv.gettimeout(), 0.0)
+        self.serv.setblocking(_testcapi.UINT_MAX + 1)
+        self.assertIsNone(self.serv.gettimeout())
+
+    _testSetBlocking_overflow = support.cpython_only(_testSetBlocking)
+
     @unittest.skipUnless(hasattr(socket, 'SOCK_NONBLOCK'),
                          'test needs socket.SOCK_NONBLOCK')
     @support.requires_linux_version(2, 6, 28)
diff --git a/Lib/test/test_structmembers.py b/Lib/test/test_structmembers.py
index 18fecc3..1c931ae 100644
--- a/Lib/test/test_structmembers.py
+++ b/Lib/test/test_structmembers.py
@@ -1,3 +1,8 @@
+import unittest
+from test import support
+
+# Skip this test if the _testcapi module isn't available.
+support.import_module('_testcapi')
 from _testcapi import _test_structmembersType, \
     CHAR_MAX, CHAR_MIN, UCHAR_MAX, \
     SHRT_MAX, SHRT_MIN, USHRT_MAX, \
@@ -6,9 +11,6 @@
     LLONG_MAX, LLONG_MIN, ULLONG_MAX, \
     PY_SSIZE_T_MAX, PY_SSIZE_T_MIN
 
-import unittest
-from test import support
-
 ts=_test_structmembersType(False,  # T_BOOL
                           1,      # T_BYTE
                           2,      # T_UBYTE
diff --git a/Lib/test/test_sys.py b/Lib/test/test_sys.py
index acf6d36..c1523db 100644
--- a/Lib/test/test_sys.py
+++ b/Lib/test/test_sys.py
@@ -610,6 +610,7 @@
         ret, out, err = assert_python_ok(*args)
         self.assertIn(b"free PyDictObjects", err)
 
+@test.support.cpython_only
 class SizeofTest(unittest.TestCase):
 
     def setUp(self):
diff --git a/Lib/test/test_tcl.py b/Lib/test/test_tcl.py
index 3693dfe..dda0c7a 100644
--- a/Lib/test/test_tcl.py
+++ b/Lib/test/test_tcl.py
@@ -1,7 +1,6 @@
 import unittest
 import sys
 import os
-import _testcapi
 from test import support
 
 # Skip this test if the _tkinter module wasn't built.
@@ -13,6 +12,11 @@
 from tkinter import Tcl
 from _tkinter import TclError
 
+try:
+    from _testcapi import INT_MAX, PY_SSIZE_T_MAX
+except ImportError:
+    INT_MAX = PY_SSIZE_T_MAX = sys.maxsize
+
 tcl_version = _tkinter.TCL_VERSION.split('.')
 try:
     for i in range(len(tcl_version)):
@@ -539,9 +543,9 @@
     def setUp(self):
         self.interp = Tcl()
 
-    @unittest.skipUnless(_testcapi.INT_MAX < _testcapi.PY_SSIZE_T_MAX,
-                         "needs UINT_MAX < SIZE_MAX")
-    @support.bigmemtest(size=_testcapi.INT_MAX + 1, memuse=5, dry_run=False)
+    @support.cpython_only
+    @unittest.skipUnless(INT_MAX < PY_SSIZE_T_MAX, "needs UINT_MAX < SIZE_MAX")
+    @support.bigmemtest(size=INT_MAX + 1, memuse=5, dry_run=False)
     def test_huge_string(self, size):
         value = ' ' * size
         self.assertRaises(OverflowError, self.interp.call, 'set', '_', value)
diff --git a/Lib/test/test_threading.py b/Lib/test/test_threading.py
index fee48a3..11c8979 100644
--- a/Lib/test/test_threading.py
+++ b/Lib/test/test_threading.py
@@ -3,7 +3,7 @@
 """
 
 import test.support
-from test.support import verbose, strip_python_stderr, import_module
+from test.support import verbose, strip_python_stderr, import_module, cpython_only
 from test.script_helper import assert_python_ok
 
 import random
@@ -773,6 +773,7 @@
         for t in threads:
             t.join()
 
+    @cpython_only
     @unittest.skipIf(_testcapi is None, "need _testcapi module")
     def test_frame_tstate_tracing(self):
         # Issue #14432: Crash when a generator is created in a C thread that is
diff --git a/Lib/test/test_time.py b/Lib/test/test_time.py
index ae3113a..1a4d873 100644
--- a/Lib/test/test_time.py
+++ b/Lib/test/test_time.py
@@ -573,6 +573,7 @@
             -(2.0 ** 100.0), 2.0 ** 100.0,
         )
 
+    @support.cpython_only
     def test_time_t(self):
         from _testcapi import pytime_object_to_time_t
         for obj, time_t in (
@@ -588,6 +589,7 @@
         for invalid in self.invalid_values:
             self.assertRaises(OverflowError, pytime_object_to_time_t, invalid)
 
+    @support.cpython_only
     def test_timeval(self):
         from _testcapi import pytime_object_to_timeval
         for obj, timeval in (
@@ -607,6 +609,7 @@
         for invalid in self.invalid_values:
             self.assertRaises(OverflowError, pytime_object_to_timeval, invalid)
 
+    @support.cpython_only
     def test_timespec(self):
         from _testcapi import pytime_object_to_timespec
         for obj, timespec in (
diff --git a/Lib/test/test_traceback.py b/Lib/test/test_traceback.py
index 9af3b92..c38c65b 100644
--- a/Lib/test/test_traceback.py
+++ b/Lib/test/test_traceback.py
@@ -1,12 +1,11 @@
 """Test cases for traceback module"""
 
-from _testcapi import traceback_print, exception_print
 from io import StringIO
 import sys
 import unittest
 import re
 from test.support import run_unittest, Error, captured_output
-from test.support import TESTFN, unlink
+from test.support import TESTFN, unlink, cpython_only
 
 import traceback
 
@@ -173,7 +172,9 @@
 
 class TracebackFormatTests(unittest.TestCase):
 
+    @cpython_only
     def test_traceback_format(self):
+        from _testcapi import traceback_print
         try:
             raise KeyError('blah')
         except KeyError:
@@ -360,7 +361,9 @@
     # This checks built-in reporting by the interpreter.
     #
 
+    @cpython_only
     def get_report(self, e):
+        from _testcapi import exception_print
         e = self.get_exception(e)
         with captured_output("stderr") as s:
             exception_print(e)
diff --git a/Lib/test/test_ucn.py b/Lib/test/test_ucn.py
index 2e63745..de71680 100644
--- a/Lib/test/test_ucn.py
+++ b/Lib/test/test_ucn.py
@@ -9,12 +9,16 @@
 
 import unittest
 import unicodedata
-import _testcapi
 
 from test import support
 from http.client import HTTPException
 from test.test_normalization import check_version
 
+try:
+    from _testcapi import INT_MAX, PY_SSIZE_T_MAX, UINT_MAX
+except ImportError:
+    INT_MAX = PY_SSIZE_T_MAX = UINT_MAX = 2**64 - 1
+
 class UnicodeNamesTest(unittest.TestCase):
 
     def checkletter(self, name, code):
@@ -216,15 +220,13 @@
             str, b"\\NSPACE", 'unicode-escape', 'strict'
         )
 
-    @unittest.skipUnless(_testcapi.INT_MAX < _testcapi.PY_SSIZE_T_MAX,
-                         "needs UINT_MAX < SIZE_MAX")
-    @support.bigmemtest(size=_testcapi.UINT_MAX + 1,
-                        memuse=2 + 1, dry_run=False)
+    @support.cpython_only
+    @unittest.skipUnless(INT_MAX < PY_SSIZE_T_MAX, "needs UINT_MAX < SIZE_MAX")
+    @support.bigmemtest(size=UINT_MAX + 1, memuse=2 + 1, dry_run=False)
     def test_issue16335(self, size):
         # very very long bogus character name
-        x = b'\\N{SPACE' + b'x' * (_testcapi.UINT_MAX + 1) + b'}'
-        self.assertEqual(len(x), len(b'\\N{SPACE}') +
-                                    (_testcapi.UINT_MAX + 1))
+        x = b'\\N{SPACE' + b'x' * (UINT_MAX + 1) + b'}'
+        self.assertEqual(len(x), len(b'\\N{SPACE}') + (UINT_MAX + 1))
         self.assertRaisesRegex(UnicodeError,
             'unknown Unicode character name',
             x.decode, 'unicode-escape'
diff --git a/Lib/test/test_unicode.py b/Lib/test/test_unicode.py
index f6b2a3f..c2ede07 100644
--- a/Lib/test/test_unicode.py
+++ b/Lib/test/test_unicode.py
@@ -1108,8 +1108,13 @@
         self.assertEqual('%.1s' % "a\xe9\u20ac", 'a')
         self.assertEqual('%.2s' % "a\xe9\u20ac", 'a\xe9')
 
-    @support.cpython_only
     def test_formatting_huge_precision(self):
+        format_string = "%.{}f".format(sys.maxsize + 1)
+        with self.assertRaises(ValueError):
+            result = format_string % 2.34
+
+    @support.cpython_only
+    def test_formatting_huge_precision_c_limits(self):
         from _testcapi import INT_MAX
         format_string = "%.{}f".format(INT_MAX + 1)
         with self.assertRaises(ValueError):
@@ -2090,6 +2095,7 @@
         self.assertEqual(PyUnicode_FromFormat(b'%.%s', b'abc'), '%.%s')
 
     # Test PyUnicode_AsWideChar()
+    @support.cpython_only
     def test_aswidechar(self):
         from _testcapi import unicode_aswidechar
         support.import_module('ctypes')
@@ -2127,6 +2133,7 @@
         self.assertEqual(wchar, nonbmp + '\0')
 
     # Test PyUnicode_AsWideCharString()
+    @support.cpython_only
     def test_aswidecharstring(self):
         from _testcapi import unicode_aswidecharstring
         support.import_module('ctypes')
@@ -2161,6 +2168,7 @@
         s += "4"
         self.assertEqual(s, "3")
 
+    @support.cpython_only
     def test_encode_decimal(self):
         from _testcapi import unicode_encodedecimal
         self.assertEqual(unicode_encodedecimal('123'),
@@ -2176,6 +2184,7 @@
             "^'decimal' codec can't encode character",
             unicode_encodedecimal, "123\u20ac", "replace")
 
+    @support.cpython_only
     def test_transform_decimal(self):
         from _testcapi import unicode_transformdecimaltoascii as transform_decimal
         self.assertEqual(transform_decimal('123'),
diff --git a/Misc/NEWS b/Misc/NEWS
index 074b2db..fb4a368 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -324,6 +324,8 @@
 Tests
 -----
 
+- Issue #20532: Tests which use _testcapi are now marked as CPython only.
+
 - Issue #19920: Added tests for TarFile.list().  Based on patch by Vajrasky Kok.
 
 - Issue #19990: Added tests for the imghdr module.  Based on patch by