diff --git a/Lib/test/string_tests.py b/Lib/test/string_tests.py
index 049929c..43ecf2a 100644
--- a/Lib/test/string_tests.py
+++ b/Lib/test/string_tests.py
@@ -28,6 +28,11 @@
     # Change in subclasses to change the behaviour of fixtesttype()
     type2test = None
 
+    # Whether the "contained items" of the container are integers in
+    # range(0, 256) (i.e. bytes, bytearray) or strings of length 1
+    # (str)
+    contains_bytes = False
+
     # All tests pass their arguments to the testing methods
     # as str objects. fixtesttype() can be used to propagate
     # these arguments to the appropriate type
@@ -117,7 +122,11 @@
         self.checkequal(0, '', 'count', 'xx', sys.maxsize, 0)
 
         self.checkraises(TypeError, 'hello', 'count')
-        self.checkraises(TypeError, 'hello', 'count', 42)
+
+        if self.contains_bytes:
+            self.checkequal(0, 'hello', 'count', 42)
+        else:
+            self.checkraises(TypeError, 'hello', 'count', 42)
 
         # For a variety of combinations,
         #    verify that str.count() matches an equivalent function
@@ -163,7 +172,11 @@
         self.checkequal( 2, 'rrarrrrrrrrra', 'find', 'a', None, 6)
 
         self.checkraises(TypeError, 'hello', 'find')
-        self.checkraises(TypeError, 'hello', 'find', 42)
+
+        if self.contains_bytes:
+            self.checkequal(-1, 'hello', 'find', 42)
+        else:
+            self.checkraises(TypeError, 'hello', 'find', 42)
 
         self.checkequal(0, '', 'find', '')
         self.checkequal(-1, '', 'find', '', 1, 1)
@@ -217,7 +230,11 @@
         self.checkequal( 2, 'rrarrrrrrrrra', 'rfind', 'a', None, 6)
 
         self.checkraises(TypeError, 'hello', 'rfind')
-        self.checkraises(TypeError, 'hello', 'rfind', 42)
+
+        if self.contains_bytes:
+            self.checkequal(-1, 'hello', 'rfind', 42)
+        else:
+            self.checkraises(TypeError, 'hello', 'rfind', 42)
 
         # For a variety of combinations,
         #    verify that str.rfind() matches __contains__
@@ -264,7 +281,11 @@
         self.checkequal( 2, 'rrarrrrrrrrra', 'index', 'a', None, 6)
 
         self.checkraises(TypeError, 'hello', 'index')
-        self.checkraises(TypeError, 'hello', 'index', 42)
+
+        if self.contains_bytes:
+            self.checkraises(ValueError, 'hello', 'index', 42)
+        else:
+            self.checkraises(TypeError, 'hello', 'index', 42)
 
     def test_rindex(self):
         self.checkequal(12, 'abcdefghiabc', 'rindex', '')
@@ -286,7 +307,11 @@
         self.checkequal( 2, 'rrarrrrrrrrra', 'rindex', 'a', None, 6)
 
         self.checkraises(TypeError, 'hello', 'rindex')
-        self.checkraises(TypeError, 'hello', 'rindex', 42)
+
+        if self.contains_bytes:
+            self.checkraises(ValueError, 'hello', 'rindex', 42)
+        else:
+            self.checkraises(TypeError, 'hello', 'rindex', 42)
 
     def test_lower(self):
         self.checkequal('hello', 'HeLLo', 'lower')
diff --git a/Lib/test/test_bytes.py b/Lib/test/test_bytes.py
index d32a44b..7acfde5 100644
--- a/Lib/test/test_bytes.py
+++ b/Lib/test/test_bytes.py
@@ -293,10 +293,27 @@
 
     def test_count(self):
         b = self.type2test(b'mississippi')
+        i = 105
+        p = 112
+        w = 119
+
         self.assertEqual(b.count(b'i'), 4)
         self.assertEqual(b.count(b'ss'), 2)
         self.assertEqual(b.count(b'w'), 0)
 
+        self.assertEqual(b.count(i), 4)
+        self.assertEqual(b.count(w), 0)
+
+        self.assertEqual(b.count(b'i', 6), 2)
+        self.assertEqual(b.count(b'p', 6), 2)
+        self.assertEqual(b.count(b'i', 1, 3), 1)
+        self.assertEqual(b.count(b'p', 7, 9), 1)
+
+        self.assertEqual(b.count(i, 6), 2)
+        self.assertEqual(b.count(p, 6), 2)
+        self.assertEqual(b.count(i, 1, 3), 1)
+        self.assertEqual(b.count(p, 7, 9), 1)
+
     def test_startswith(self):
         b = self.type2test(b'hello')
         self.assertFalse(self.type2test().startswith(b"anything"))
@@ -327,35 +344,81 @@
 
     def test_find(self):
         b = self.type2test(b'mississippi')
+        i = 105
+        w = 119
+
         self.assertEqual(b.find(b'ss'), 2)
-        self.assertEqual(b.find(b'ss', 3), 5)
-        self.assertEqual(b.find(b'ss', 1, 7), 2)
-        self.assertEqual(b.find(b'ss', 1, 3), -1)
         self.assertEqual(b.find(b'w'), -1)
         self.assertEqual(b.find(b'mississippian'), -1)
 
+        self.assertEqual(b.find(i), 1)
+        self.assertEqual(b.find(w), -1)
+
+        self.assertEqual(b.find(b'ss', 3), 5)
+        self.assertEqual(b.find(b'ss', 1, 7), 2)
+        self.assertEqual(b.find(b'ss', 1, 3), -1)
+
+        self.assertEqual(b.find(i, 6), 7)
+        self.assertEqual(b.find(i, 1, 3), 1)
+        self.assertEqual(b.find(w, 1, 3), -1)
+
     def test_rfind(self):
         b = self.type2test(b'mississippi')
+        i = 105
+        w = 119
+
         self.assertEqual(b.rfind(b'ss'), 5)
-        self.assertEqual(b.rfind(b'ss', 3), 5)
-        self.assertEqual(b.rfind(b'ss', 0, 6), 2)
         self.assertEqual(b.rfind(b'w'), -1)
         self.assertEqual(b.rfind(b'mississippian'), -1)
 
+        self.assertEqual(b.rfind(i), 10)
+        self.assertEqual(b.rfind(w), -1)
+
+        self.assertEqual(b.rfind(b'ss', 3), 5)
+        self.assertEqual(b.rfind(b'ss', 0, 6), 2)
+
+        self.assertEqual(b.rfind(i, 1, 3), 1)
+        self.assertEqual(b.rfind(i, 3, 9), 7)
+        self.assertEqual(b.rfind(w, 1, 3), -1)
+
     def test_index(self):
-        b = self.type2test(b'world')
-        self.assertEqual(b.index(b'w'), 0)
-        self.assertEqual(b.index(b'orl'), 1)
-        self.assertRaises(ValueError, b.index, b'worm')
-        self.assertRaises(ValueError, b.index, b'ldo')
+        b = self.type2test(b'mississippi')
+        i = 105
+        w = 119
+
+        self.assertEqual(b.index(b'ss'), 2)
+        self.assertRaises(ValueError, b.index, b'w')
+        self.assertRaises(ValueError, b.index, b'mississippian')
+
+        self.assertEqual(b.index(i), 1)
+        self.assertRaises(ValueError, b.index, w)
+
+        self.assertEqual(b.index(b'ss', 3), 5)
+        self.assertEqual(b.index(b'ss', 1, 7), 2)
+        self.assertRaises(ValueError, b.index, b'ss', 1, 3)
+
+        self.assertEqual(b.index(i, 6), 7)
+        self.assertEqual(b.index(i, 1, 3), 1)
+        self.assertRaises(ValueError, b.index, w, 1, 3)
 
     def test_rindex(self):
-        # XXX could be more rigorous
-        b = self.type2test(b'world')
-        self.assertEqual(b.rindex(b'w'), 0)
-        self.assertEqual(b.rindex(b'orl'), 1)
-        self.assertRaises(ValueError, b.rindex, b'worm')
-        self.assertRaises(ValueError, b.rindex, b'ldo')
+        b = self.type2test(b'mississippi')
+        i = 105
+        w = 119
+
+        self.assertEqual(b.rindex(b'ss'), 5)
+        self.assertRaises(ValueError, b.rindex, b'w')
+        self.assertRaises(ValueError, b.rindex, b'mississippian')
+
+        self.assertEqual(b.rindex(i), 10)
+        self.assertRaises(ValueError, b.rindex, w)
+
+        self.assertEqual(b.rindex(b'ss', 3), 5)
+        self.assertEqual(b.rindex(b'ss', 0, 6), 2)
+
+        self.assertEqual(b.rindex(i, 1, 3), 1)
+        self.assertEqual(b.rindex(i, 3, 9), 7)
+        self.assertRaises(ValueError, b.rindex, w, 1, 3)
 
     def test_replace(self):
         b = self.type2test(b'mississippi')
@@ -552,6 +615,14 @@
         self.assertEqual(True, b.startswith(h, None, -2))
         self.assertEqual(False, b.startswith(x, None, None))
 
+    def test_integer_arguments_out_of_byte_range(self):
+        b = self.type2test(b'hello')
+
+        for method in (b.count, b.find, b.index, b.rfind, b.rindex):
+            self.assertRaises(ValueError, method, -1)
+            self.assertRaises(ValueError, method, 256)
+            self.assertRaises(ValueError, method, 9999)
+
     def test_find_etc_raise_correct_error_messages(self):
         # issue 11828
         b = self.type2test(b'hello')
@@ -1161,9 +1232,11 @@
 
 class ByteArrayAsStringTest(FixedStringTest):
     type2test = bytearray
+    contains_bytes = True
 
 class BytesAsStringTest(FixedStringTest):
     type2test = bytes
+    contains_bytes = True
 
 
 class SubclassTest(unittest.TestCase):