diff --git a/OpenSSL/_util.py b/OpenSSL/_util.py
index baeecc6..ed5ddce 100644
--- a/OpenSSL/_util.py
+++ b/OpenSSL/_util.py
@@ -1,3 +1,5 @@
+import sys
+
 from six import PY3, binary_type, text_type
 
 from cryptography.hazmat.bindings.openssl.binding import Binding
@@ -45,6 +47,23 @@
 
 
 
+def path_string(s):
+    """
+    Convert a Python string to a :py:class:`bytes` string identifying the same
+    path and which can be passed into an OpenSSL API accepting a filename.
+
+    :param s: An instance of :py:class:`bytes` or :py:class:`unicode`.
+
+    :return: An instance of :py:class:`bytes`.
+    """
+    if isinstance(s, binary_type):
+        return s
+    elif isinstance(s, text_type):
+        return s.encode(sys.getfilesystemencoding())
+    else:
+        raise TypeError("Path must be represented as bytes or unicode string")
+
+
 if PY3:
     def byte_string(s):
         return s.encode("charmap")
diff --git a/OpenSSL/rand.py b/OpenSSL/rand.py
index 311c00b..7592fe0 100644
--- a/OpenSSL/rand.py
+++ b/OpenSSL/rand.py
@@ -11,7 +11,8 @@
 from OpenSSL._util import (
     ffi as _ffi,
     lib as _lib,
-    exception_from_error_queue as _exception_from_error_queue)
+    exception_from_error_queue as _exception_from_error_queue,
+    path_string as _path_string)
 
 
 class Error(Exception):
@@ -136,8 +137,7 @@
                      to read the entire file
     :return: The number of bytes read
     """
-    if not isinstance(filename, (str, _builtin_bytes)):
-        raise TypeError("filename must be a string")
+    filename = _path_string(filename)
 
     if maxbytes is _unspecified:
         maxbytes = -1
@@ -155,9 +155,7 @@
     :param filename: The file to write data to
     :return: The number of bytes written
     """
-    if not isinstance(filename, (str, _builtin_bytes)):
-        raise TypeError("filename must be a string")
-
+    filename = _path_string(filename)
     return _lib.RAND_write_file(filename)
 
 
diff --git a/OpenSSL/test/test_rand.py b/OpenSSL/test/test_rand.py
index fbb452c..c6f985a 100644
--- a/OpenSSL/test/test_rand.py
+++ b/OpenSSL/test/test_rand.py
@@ -176,26 +176,52 @@
         self.assertRaises(TypeError, rand.write_file, None)
         self.assertRaises(TypeError, rand.write_file, "foo", None)
 
-    def test_files(self):
+    def _read_write_test(self, path):
         """
-        Test reading and writing of files via rand functions.
+        Verify that ``rand.write_file`` and ``rand.load_file`` can be used.
         """
-        # Write random bytes to a file
-        tmpfile = self.mktemp()
-        # Make sure it exists (so cleanup definitely succeeds)
-        fObj = open(tmpfile, 'w')
-        fObj.close()
+        # Create the file so cleanup is more straightforward
+        with open(path, "w"):
+            pass
+
         try:
-            rand.write_file(tmpfile)
+            # Write random bytes to a file
+            rand.write_file(path)
+
             # Verify length of written file
-            size = os.stat(tmpfile)[stat.ST_SIZE]
+            size = os.stat(path)[stat.ST_SIZE]
             self.assertEqual(1024, size)
+
             # Read random bytes from file
-            rand.load_file(tmpfile)
-            rand.load_file(tmpfile, 4)  # specify a length
+            rand.load_file(path)
+            rand.load_file(path, 4)  # specify a length
         finally:
             # Cleanup
-            os.unlink(tmpfile)
+            os.unlink(path)
+
+
+    def test_bytes_paths(self):
+        """
+        Random data can be saved and loaded to files with paths specified as
+        bytes.
+        """
+        path = self.mktemp()
+        # This is the UTF-8 encoding of the SNOWMAN unicode code point.
+        path += b("\xe2\x98\x83").decode("utf-8").encode(
+            sys.getfilesystemencoding()
+        )
+        self._read_write_test(path)
+
+
+    def test_unicode_paths(self):
+        """
+        Random data can be saved and loaded to files with paths specified as
+        unicode.
+        """
+        path = self.mktemp().decode('utf-8')
+        # This is the UTF-8 encoding of the SNOWMAN unicode code point.
+        path += b("\xe2\x98\x83").decode('utf-8')
+        self._read_write_test(path)
 
 
 if __name__ == '__main__':