diff --git a/.github/workflows/testsuite.yml b/.github/workflows/testsuite.yml
index 33aee2b..92e7c4d 100644
--- a/.github/workflows/testsuite.yml
+++ b/.github/workflows/testsuite.yml
@@ -29,6 +29,10 @@
         include:
           - python-version: "pypy-3.7"
             os: ubuntu-latest
+          - python-version: "pypy-3.9"
+            os: ubuntu-latest
+          - python-version: "pypy-3.10"
+            os: ubuntu-latest
 
     steps:
     - uses: actions/checkout@v3
diff --git a/CHANGES.md b/CHANGES.md
index 4953808..15739ac 100644
--- a/CHANGES.md
+++ b/CHANGES.md
@@ -11,6 +11,7 @@
 * Adapt to changes in Python 3.12 beta1 (only working partially,
   see [#830](../../issues/830) and [#831](../../issues/831)).
 * Adapt to changes in `shutil` in Python 3.12 beta2 (see [#814](../../issues/814)).
+* Fix support for newer PyPi versions (see [#859](../../issues/859)).
 
 ### Documentation
 * Added a note regarding the incompatibility of the built-in `sqlite3` module with
@@ -18,6 +19,7 @@
 
 ### Infrastructure
 * Added pytype check for non-test modules in CI (see [#599](../../issues/599)).
+* Added tests for different pypy3 versions.
 
 ## [Version 5.2.2](https://pypi.python.org/pypi/pyfakefs/5.2.2) (2023-04-13)
 Fixes a regression in 5.2.0
diff --git a/pyfakefs/fake_io.py b/pyfakefs/fake_io.py
index 2f78230..fc719ad 100644
--- a/pyfakefs/fake_io.py
+++ b/pyfakefs/fake_io.py
@@ -33,6 +33,7 @@
 
 from pyfakefs.fake_file import AnyFileWrapper
 from pyfakefs.fake_open import FakeFileOpen
+from pyfakefs.helpers import IS_PYPY
 
 if TYPE_CHECKING:
     from pyfakefs.fake_filesystem import FakeFilesystem
@@ -124,7 +125,7 @@
             function may be overridden by an earlier call to the
             PyFile_SetOpenCodeHook(). This behavior is not reproduced here.
             """
-            if not isinstance(path, str):
+            if not isinstance(path, str) and not IS_PYPY:
                 raise TypeError("open_code() argument 'path' must be str, not int")
             patch_mode = self.filesystem.patch_open_code
             if (
diff --git a/pyfakefs/fake_os.py b/pyfakefs/fake_os.py
index 989d5c9..ba19bf8 100644
--- a/pyfakefs/fake_os.py
+++ b/pyfakefs/fake_os.py
@@ -1348,7 +1348,7 @@
 
         @functools.wraps(f)
         def wrapped(*args, **kwargs):
-            if not f.__name__.startswith("_") and FakeOsModule.use_original:
+            if FakeOsModule.use_original:
                 # remove the `self` argument for FakeOsModule methods
                 if args and isinstance(args[0], FakeOsModule):
                     args = args[1:]
@@ -1358,7 +1358,8 @@
         return wrapped
 
     for name, fn in inspect.getmembers(FakeOsModule, inspect.isfunction):
-        setattr(FakeOsModule, name, handle_original_call(fn))
+        if not fn.__name__.startswith("_"):
+            setattr(FakeOsModule, name, handle_original_call(fn))
 
 
 @contextmanager
diff --git a/pyfakefs/fake_pathlib.py b/pyfakefs/fake_pathlib.py
index 17c3ebb..6e096d1 100644
--- a/pyfakefs/fake_pathlib.py
+++ b/pyfakefs/fake_pathlib.py
@@ -47,6 +47,7 @@
 from pyfakefs.fake_filesystem import FakeFilesystem
 from pyfakefs.fake_open import FakeFileOpen
 from pyfakefs.fake_os import FakeOsModule, use_original_os
+from pyfakefs.helpers import IS_PYPY
 
 
 def init_module(filesystem):
@@ -130,9 +131,8 @@
                     "chmod() got an unexpected keyword " "argument 'follow_symlinks'"
                 )
 
-            if (
-                not kwargs["follow_symlinks"]
-                and os.chmod not in os.supports_follow_symlinks
+            if not kwargs["follow_symlinks"] and (
+                os.chmod not in os.supports_follow_symlinks or IS_PYPY
             ):
                 raise NotImplementedError(
                     "`follow_symlinks` for chmod() is not available " "on this system"
@@ -916,7 +916,8 @@
         return wrapped
 
     for name, fn in inspect.getmembers(RealPath, inspect.isfunction):
-        setattr(RealPath, name, with_original_os(fn))
+        if not name.startswith("__"):
+            setattr(RealPath, name, with_original_os(fn))
 
 
 class RealPathlibPathModule:
diff --git a/pyfakefs/pytest_plugin.py b/pyfakefs/pytest_plugin.py
index 0ecdd0e..fed7a55 100644
--- a/pyfakefs/pytest_plugin.py
+++ b/pyfakefs/pytest_plugin.py
@@ -10,6 +10,7 @@
 """
 import py
 import pytest
+from _pytest import capture
 
 from pyfakefs.fake_filesystem_unittest import Patcher
 
@@ -20,6 +21,7 @@
 
 Patcher.SKIPMODULES.add(py)
 Patcher.SKIPMODULES.add(pytest)
+Patcher.SKIPMODULES.add(capture)
 if pathlib is not None:
     Patcher.SKIPMODULES.add(pathlib)
 
diff --git a/pyfakefs/tests/fake_filesystem_shutil_test.py b/pyfakefs/tests/fake_filesystem_shutil_test.py
index a3b4ae3..83d4672 100644
--- a/pyfakefs/tests/fake_filesystem_shutil_test.py
+++ b/pyfakefs/tests/fake_filesystem_shutil_test.py
@@ -25,7 +25,7 @@
 from pathlib import Path
 
 from pyfakefs import fake_filesystem_unittest
-from pyfakefs.helpers import get_uid, set_uid, is_root
+from pyfakefs.helpers import get_uid, set_uid, is_root, IS_PYPY
 from pyfakefs.tests.test_utils import RealFsTestMixin
 
 is_windows = sys.platform == "win32"
@@ -231,6 +231,7 @@
         self.assertAlmostEqual(src_stat.st_atime, dst_stat.st_atime, places=2)
         self.assertAlmostEqual(src_stat.st_mtime, dst_stat.st_mtime, places=2)
 
+    @unittest.skipIf(IS_PYPY, "Functionality not supported in PyPy")
     def test_copystat_symlinks(self):
         """Regression test for #799"""
         self.skip_if_symlink_not_supported()
diff --git a/pyfakefs/tests/fake_open_test.py b/pyfakefs/tests/fake_open_test.py
index cdb8a7b..21a5043 100644
--- a/pyfakefs/tests/fake_open_test.py
+++ b/pyfakefs/tests/fake_open_test.py
@@ -25,7 +25,7 @@
 import unittest
 
 from pyfakefs import fake_filesystem, helpers
-from pyfakefs.helpers import is_root
+from pyfakefs.helpers import is_root, IS_PYPY
 from pyfakefs.fake_io import FakeIoModule
 from pyfakefs.fake_filesystem_unittest import PatchMode
 from pyfakefs.tests.test_utils import RealFsTestCase
@@ -1057,10 +1057,19 @@
             self.filesystem.patch_open_code = False
         super(FakeFilePatchedOpenCodeTest, self).tearDown()
 
+    @unittest.skipIf(IS_PYPY, "Different behavior in PyPy")
     def test_invalid_path(self):
         with self.assertRaises(TypeError):
             self.open_code(4)
 
+    @unittest.skipIf(not IS_PYPY, "Different behavior in PyPy")
+    def test_open_code_fd_pypy(self):
+        file_path = self.make_path("foo")
+        self.create_file(file_path, contents="test")
+        fd = self.os.open(file_path, os.O_RDONLY)
+        with self.open_code(fd) as f:
+            assert f.read() == b"test"
+
     def test_byte_contents_open_code(self):
         byte_fractions = b"\xe2\x85\x93 \xe2\x85\x94 \xe2\x85\x95 \xe2\x85\x96"
         file_path = self.make_path("foo")
@@ -1090,6 +1099,7 @@
         else:
             self.open_code = self.fake_io_module.open_code
 
+    @unittest.skipIf(IS_PYPY, "Different behavior in PyPy")
     def test_invalid_path(self):
         with self.assertRaises(TypeError):
             self.open_code(4)
diff --git a/pyfakefs/tests/patched_packages_test.py b/pyfakefs/tests/patched_packages_test.py
index f57bb58..1c76091 100644
--- a/pyfakefs/tests/patched_packages_test.py
+++ b/pyfakefs/tests/patched_packages_test.py
@@ -15,6 +15,7 @@
 with pyfakefs.
 """
 import os
+import sys
 import unittest
 
 from pyfakefs import fake_filesystem_unittest
@@ -36,7 +37,9 @@
     openpyxl = None
 
 
-@unittest.skipIf(IS_PYPY, "Has a problem with current PyPy")
+@unittest.skipIf(
+    IS_PYPY and sys.version_info < (3, 8), "Has a problem with older PyPy versions"
+)
 class TestPatchedPackages(fake_filesystem_unittest.TestCase):
     def setUp(self):
         self.setUpPyfakefs()