diff --git a/CHANGELOG.md b/CHANGELOG.md
index 99a3e73b8e2..f941c75a0ea 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -5,6 +5,17 @@ PyO3 versions, please see the [migration guide](https://pyo3.rs/master/migration
 The format is based on [Keep a Changelog](http://keepachangelog.com/en/1.0.0/)
 and this project adheres to [Semantic Versioning](http://semver.org/spec/v2.0.0.html).
 
+## Unreleased
+### Added
+- Add FFI definition `PyCFunction_CheckExact` for Python 3.9 and later. [#1425](https://github.com/PyO3/pyo3/pull/1425)
+
+### Changed
+- Deprecate FFI definition `PyCFunction_Call` for Python 3.9 and later. [#1425](https://github.com/PyO3/pyo3/pull/1425)
+- Deprecate FFI definitions `PyModule_GetFilename`, `PyMethodDef_INIT`. [#1425](https://github.com/PyO3/pyo3/pull/1425)
+
+### Fixed
+- Remove FFI definition `PyCFunction_ClearFreeList` for Python 3.9 and later. [#1425](https://github.com/PyO3/pyo3/pull/1425)
+
 ## [0.13.2] - 2021-02-12
 ### Packaging
 - Lower minimum supported Rust version to 1.41. [#1421](https://github.com/PyO3/pyo3/pull/1421)
diff --git a/src/derive_utils.rs b/src/derive_utils.rs
index cd0f5bde1ff..a6c902fb579 100644
--- a/src/derive_utils.rs
+++ b/src/derive_utils.rs
@@ -136,6 +136,7 @@ impl ModuleDef {
     /// # Safety
     /// `name` must be a null-terminated string.
     pub const unsafe fn new(name: &'static str) -> Self {
+        #[allow(deprecated)]
         let mut init = ffi::PyModuleDef_INIT;
         init.m_name = name.as_ptr() as *const _;
         ModuleDef(UnsafeCell::new(init))
diff --git a/src/ffi/methodobject.rs b/src/ffi/methodobject.rs
index fdf1797ea2a..ecfbc81c205 100644
--- a/src/ffi/methodobject.rs
+++ b/src/ffi/methodobject.rs
@@ -1,4 +1,6 @@
 use crate::ffi::object::{PyObject, PyTypeObject, Py_TYPE};
+#[cfg(Py_3_9)]
+use crate::ffi::PyObject_TypeCheck;
 use std::mem;
 use std::os::raw::{c_char, c_int};
 
@@ -8,6 +10,19 @@ extern "C" {
     pub static mut PyCFunction_Type: PyTypeObject;
 }
 
+#[cfg(Py_3_9)]
+#[inline]
+pub unsafe fn PyCFunction_CheckExact(op: *mut PyObject) -> c_int {
+    (Py_TYPE(op) == &mut PyCFunction_Type) as c_int
+}
+
+#[cfg(Py_3_9)]
+#[inline]
+pub unsafe fn PyCFunction_Check(op: *mut PyObject) -> c_int {
+    PyObject_TypeCheck(op, &mut PyCFunction_Type)
+}
+
+#[cfg(not(Py_3_9))]
 #[inline]
 pub unsafe fn PyCFunction_Check(op: *mut PyObject) -> c_int {
     (Py_TYPE(op) == &mut PyCFunction_Type) as c_int
@@ -38,11 +53,14 @@ pub type _PyCFunctionFastWithKeywords = unsafe extern "C" fn(
     kwnames: *mut PyObject,
 ) -> *mut PyObject;
 
+// skipped PyCMethod (since 3.9)
+
 extern "C" {
     #[cfg_attr(PyPy, link_name = "PyPyCFunction_GetFunction")]
     pub fn PyCFunction_GetFunction(f: *mut PyObject) -> Option<PyCFunction>;
     pub fn PyCFunction_GetSelf(f: *mut PyObject) -> *mut PyObject;
     pub fn PyCFunction_GetFlags(f: *mut PyObject) -> c_int;
+    #[cfg_attr(Py_3_9, deprecated(note = "Python 3.9"))]
     pub fn PyCFunction_Call(
         f: *mut PyObject,
         args: *mut PyObject,
@@ -59,6 +77,10 @@ pub struct PyMethodDef {
     pub ml_doc: *const c_char,
 }
 
+/// Helper initial value of [`PyMethodDef`] for a Python class.
+///
+/// Not present in the Python C API.
+#[deprecated(note = "not present in Python headers; to be removed")]
 pub const PyMethodDef_INIT: PyMethodDef = PyMethodDef {
     ml_name: std::ptr::null(),
     ml_meth: None,
@@ -73,17 +95,19 @@ impl Default for PyMethodDef {
 }
 
 extern "C" {
+    #[cfg_attr(PyPy, link_name = "PyPyCFunction_New")]
+    pub fn PyCFunction_New(ml: *mut PyMethodDef, slf: *mut PyObject) -> *mut PyObject;
+
     #[cfg_attr(PyPy, link_name = "PyPyCFunction_NewEx")]
     pub fn PyCFunction_NewEx(
         ml: *mut PyMethodDef,
         slf: *mut PyObject,
         module: *mut PyObject,
     ) -> *mut PyObject;
-
-    #[cfg_attr(PyPy, link_name = "PyPyCFunction_New")]
-    pub fn PyCFunction_New(ml: *mut PyMethodDef, slf: *mut PyObject) -> *mut PyObject;
 }
 
+// skipped non-limited / 3.9 PyCMethod_New
+
 /* Flag passed to newmethodobject */
 pub const METH_VARARGS: c_int = 0x0001;
 pub const METH_KEYWORDS: c_int = 0x0002;
@@ -109,6 +133,10 @@ be specified alone or with METH_KEYWORDS. */
 #[cfg(all(Py_3_7, not(Py_LIMITED_API)))]
 pub const METH_FASTCALL: c_int = 0x0080;
 
+// skipped METH_STACKLESS
+// skipped METH_METHOD
+
 extern "C" {
+    #[cfg(not(Py_3_9))]
     pub fn PyCFunction_ClearFreeList() -> c_int;
 }
diff --git a/src/ffi/mod.rs b/src/ffi/mod.rs
index 855b1e2a87f..20962ea2939 100644
--- a/src/ffi/mod.rs
+++ b/src/ffi/mod.rs
@@ -127,6 +127,10 @@ mod listobject;
 mod longobject;
 pub(crate) mod marshal;
 mod memoryobject;
+mod methodobject; // TODO: incomplete
+mod modsupport; // TODO: incomplete
+mod moduleobject; // TODO: incomplete
+
 // skipped namespaceobject.h
 // skipped odictobject.h
 // skipped opcode.h
@@ -174,8 +178,6 @@ mod unicodeobject; // TODO supports PEP-384 only; needs adjustment for Python 3.
 mod rangeobject; // TODO supports PEP-384 only; needs adjustment for Python 3.3 and 3.5
 mod tupleobject; // TODO supports PEP-384 only; needs adjustment for Python 3.3 and 3.5
                  // mod odictobject; TODO new in 3.5
-mod methodobject; // TODO supports PEP-384 only; needs adjustment for Python 3.3 and 3.5
-mod moduleobject;
 mod setobject; // TODO supports PEP-384 only; needs adjustment for Python 3.3 and 3.5
                // mod classobject; TODO excluded by PEP-384
 mod pycapsule; // TODO supports PEP-384 only; needs adjustment for Python 3.3 and 3.5
@@ -191,7 +193,6 @@ mod pystate; // TODO supports PEP-384 only; needs adjustment for Python 3.3 and
 
 #[cfg(Py_LIMITED_API)]
 mod pyarena {}
-mod modsupport; // TODO supports PEP-384 only; needs adjustment for Python 3.3 and 3.5
 #[cfg(not(Py_LIMITED_API))]
 mod pyarena; // TODO: incomplete
 mod pythonrun; // TODO some functions need to be moved to pylifecycle
diff --git a/src/ffi/modsupport.rs b/src/ffi/modsupport.rs
index d2c886f0710..62826c600dc 100644
--- a/src/ffi/modsupport.rs
+++ b/src/ffi/modsupport.rs
@@ -28,12 +28,32 @@ extern "C" {
     ) -> c_int;
     #[cfg_attr(PyPy, link_name = "PyPy_BuildValue")]
     pub fn Py_BuildValue(arg1: *const c_char, ...) -> *mut PyObject;
-    #[cfg_attr(PyPy, link_name = "_PyPy_BuildValue_SizeT")]
+    // #[cfg_attr(PyPy, link_name = "_PyPy_BuildValue_SizeT")]
     //pub fn _Py_BuildValue_SizeT(arg1: *const c_char, ...)
     // -> *mut PyObject;
-    #[cfg_attr(PyPy, link_name = "PyPy_VaBuildValue")]
+    // #[cfg_attr(PyPy, link_name = "PyPy_VaBuildValue")]
+
+    // skipped non-limited _PyArg_UnpackStack
+    // skipped non-limited _PyArg_NoKeywords
+    // skipped non-limited _PyArg_NoKwnames
+    // skipped non-limited _PyArg_NoPositional
+    // skipped non-limited _PyArg_BadArgument
+    // skipped non-limited _PyArg_CheckPositional
+
     //pub fn Py_VaBuildValue(arg1: *const c_char, arg2: va_list)
     // -> *mut PyObject;
+
+    // skipped non-limited _Py_VaBuildStack
+    // skipped non-limited _PyArg_Parser
+
+    // skipped non-limited _PyArg_ParseTupleAndKeywordsFast
+    // skipped non-limited _PyArg_ParseStack
+    // skipped non-limited _PyArg_ParseStackAndKeywords
+    // skipped non-limited _PyArg_VaParseTupleAndKeywordsFast
+    // skipped non-limited _PyArg_UnpackKeywords
+    // skipped non-limited _PyArg_Fini
+
+    // skipped PyModule_AddObjectRef
     #[cfg_attr(PyPy, link_name = "PyPyModule_AddObject")]
     pub fn PyModule_AddObject(
         arg1: *mut PyObject,
@@ -49,6 +69,9 @@ extern "C" {
         arg2: *const c_char,
         arg3: *const c_char,
     ) -> c_int;
+    // skipped non-limited / 3.9 PyModule_AddType
+    // skipped PyModule_AddIntMacro
+    // skipped PyModule_AddStringMacro
     pub fn PyModule_SetDocString(arg1: *mut PyObject, arg2: *const c_char) -> c_int;
     pub fn PyModule_AddFunctions(arg1: *mut PyObject, arg2: *mut PyMethodDef) -> c_int;
     pub fn PyModule_ExecDef(module: *mut PyObject, def: *mut PyModuleDef) -> c_int;
@@ -122,3 +145,5 @@ pub unsafe fn PyModule_FromDefAndSpec(def: *mut PyModuleDef, spec: *mut PyObject
         },
     )
 }
+
+// skipped non-limited _Py_PackageContext
diff --git a/src/ffi/moduleobject.rs b/src/ffi/moduleobject.rs
index 1093d28fa8a..d43efa78e1b 100644
--- a/src/ffi/moduleobject.rs
+++ b/src/ffi/moduleobject.rs
@@ -30,8 +30,12 @@ extern "C" {
     #[cfg_attr(PyPy, link_name = "PyPyModule_GetName")]
     pub fn PyModule_GetName(arg1: *mut PyObject) -> *const c_char;
     #[cfg(not(all(windows, PyPy)))]
+    #[deprecated(note = "Python 3.2")]
     pub fn PyModule_GetFilename(arg1: *mut PyObject) -> *const c_char;
     pub fn PyModule_GetFilenameObject(arg1: *mut PyObject) -> *mut PyObject;
+    // skipped non-limited _PyModule_Clear
+    // skipped non-limited _PyModule_ClearDict
+    // skipped non-limited _PyModuleSpec_IsInitializing
     #[cfg_attr(PyPy, link_name = "PyPyModule_GetDef")]
     pub fn PyModule_GetDef(arg1: *mut PyObject) -> *mut PyModuleDef;
     #[cfg_attr(PyPy, link_name = "PyPyModule_GetState")]
@@ -71,6 +75,8 @@ pub struct PyModuleDef_Slot {
 pub const Py_mod_create: c_int = 1;
 pub const Py_mod_exec: c_int = 2;
 
+// skipped non-limited _Py_mod_LAST_SLOT
+
 #[repr(C)]
 #[derive(Copy, Clone)]
 pub struct PyModuleDef {
@@ -85,6 +91,10 @@ pub struct PyModuleDef {
     pub m_free: Option<freefunc>,
 }
 
+/// Helper initial value of [`PyModuleDef`] for a Python class.
+///
+/// Not present in the Python C API.
+#[deprecated(note = "not present in Python headers; to be removed")]
 pub const PyModuleDef_INIT: PyModuleDef = PyModuleDef {
     m_base: PyModuleDef_HEAD_INIT,
     m_name: std::ptr::null(),
diff --git a/src/pyclass.rs b/src/pyclass.rs
index 2c2fe45db7a..066ea238424 100644
--- a/src/pyclass.rs
+++ b/src/pyclass.rs
@@ -350,6 +350,7 @@ fn py_class_method_defs<T: PyMethods>() -> (
     }
 
     if !defs.is_empty() {
+        #[allow(deprecated)]
         defs.push(ffi::PyMethodDef_INIT);
     }
 
diff --git a/src/types/module.rs b/src/types/module.rs
index 11d34f1c59d..71a219a29bb 100644
--- a/src/types/module.rs
+++ b/src/types/module.rs
@@ -121,7 +121,12 @@ impl PyModule {
     /// May fail if the module does not have a `__file__` attribute.
     #[cfg(not(all(windows, PyPy)))]
     pub fn filename(&self) -> PyResult<&str> {
-        unsafe { self.str_from_ptr(ffi::PyModule_GetFilename(self.as_ptr())) }
+        use crate::types::PyString;
+        unsafe {
+            self.py()
+                .from_owned_ptr_or_err::<PyString>(ffi::PyModule_GetFilenameObject(self.as_ptr()))?
+                .to_str()
+        }
     }
 
     /// Calls a function in the module.