diff --git a/docs/source/faq.rst b/docs/source/faq.rst
index 43ba3d0d066e2..2a79498dd7929 100644
--- a/docs/source/faq.rst
+++ b/docs/source/faq.rst
@@ -201,7 +201,8 @@ the following aspects, among others:
 Does it run on PyPy?
 *********************
 
-No. MyPy relies on `typed-ast 
+Somewhat. With PyPy 3.8, mypy is at least able to type check itself.
+With older versions of PyPy, mypy relies on `typed-ast 
 <https://github.com/python/typed_ast>`_, which uses several APIs that
 PyPy does not support (including some internal CPython APIs).
 
diff --git a/mypy-requirements.txt b/mypy-requirements.txt
index 96888e2211282..50e690b71c687 100644
--- a/mypy-requirements.txt
+++ b/mypy-requirements.txt
@@ -1,4 +1,4 @@
 typing_extensions>=3.7.4
 mypy_extensions>=0.4.3,<0.5.0
-typed_ast>=1.4.0,<1.5.0
+typed_ast>=1.4.0,<1.5.0; python_version<'3.8'
 tomli>=1.1.0,<1.2.0
diff --git a/mypy/build.py b/mypy/build.py
index ba671d6ea7006..06d5ac3f72cf7 100644
--- a/mypy/build.py
+++ b/mypy/build.py
@@ -15,6 +15,7 @@
 import gc
 import json
 import os
+import platform
 import re
 import stat
 import sys
@@ -201,8 +202,9 @@ def _build(sources: List[BuildSource],
            stderr: TextIO,
            extra_plugins: Sequence[Plugin],
            ) -> BuildResult:
-    # This seems the most reasonable place to tune garbage collection.
-    gc.set_threshold(150 * 1000)
+    if platform.python_implementation() == 'CPython':
+        # This seems the most reasonable place to tune garbage collection.
+        gc.set_threshold(150 * 1000)
 
     data_dir = default_data_dir()
     fscache = fscache or FileSystemCache()
diff --git a/test-requirements.txt b/test-requirements.txt
index 9091914a5d2bf..2756dd11986fd 100644
--- a/test-requirements.txt
+++ b/test-requirements.txt
@@ -11,6 +11,7 @@ pytest-xdist>=1.34.0,<2.0.0
 pytest-forked>=1.3.0,<2.0.0
 pytest-cov>=2.10.0,<3.0.0
 py>=1.5.2
+typed_ast>=1.4.0,<1.5.0; python_version>='3.8'
 virtualenv>=20.6.0
 setuptools!=50
 importlib-metadata>=4.6.1,<5.0.0