diff --git a/sphinx/builders/latex/util.py b/sphinx/builders/latex/util.py
index b7d79121c..0e3eb739d 100644
--- a/sphinx/builders/latex/util.py
+++ b/sphinx/builders/latex/util.py
@@ -8,6 +8,8 @@
     :license: BSD, see LICENSE for details.
 """
 
+from typing import Optional
+
 from docutils.writers.latex2e import Babel
 
 
@@ -40,7 +42,7 @@ class ExtBabel(Babel):
             self.supported = False
             return 'english'  # fallback to english
 
-    def get_mainlanguage_options(self) -> str:
+    def get_mainlanguage_options(self) -> Optional[str]:
         """Return options for polyglossia's ``\\setmainlanguage``."""
         if self.use_polyglossia is False:
             return None
diff --git a/sphinx/deprecation.py b/sphinx/deprecation.py
index 5e5e673d2..a3fda02c4 100644
--- a/sphinx/deprecation.py
+++ b/sphinx/deprecation.py
@@ -73,6 +73,6 @@ class DeprecatedDict(dict):
         warnings.warn(self.message, self.warning, stacklevel=2)
         return super().get(key, default)
 
-    def update(self, other: Dict = None) -> None:  # type: ignore
+    def update(self, other: Dict) -> None:  # type: ignore
         warnings.warn(self.message, self.warning, stacklevel=2)
         super().update(other)
diff --git a/sphinx/ext/napoleon/iterators.py b/sphinx/ext/napoleon/iterators.py
index e91a3ec14..fc41afdb3 100644
--- a/sphinx/ext/napoleon/iterators.py
+++ b/sphinx/ext/napoleon/iterators.py
@@ -11,7 +11,7 @@
 """
 
 import collections
-from typing import Any, Iterable
+from typing import Any, Iterable, Optional
 
 
 class peek_iter:
@@ -62,7 +62,7 @@ class peek_iter:
     def __next__(self, n: int = None) -> Any:
         return self.next(n)
 
-    def _fillcache(self, n: int) -> None:
+    def _fillcache(self, n: Optional[int]) -> None:
         """Cache `n` items. If `n` is 0 or None, then 1 item is cached."""
         if not n:
             n = 1
@@ -123,7 +123,7 @@ class peek_iter:
             result = [self._cache.popleft() for i in range(n)]
         return result
 
-    def peek(self, n: int = None) -> Any:
+    def peek(self, n: Optional[int] = None) -> Any:
         """Preview the next item or `n` items of the iterator.
 
         The iterator is not advanced when peek is called.
@@ -220,7 +220,7 @@ class modify_iter(peek_iter):
                             'modifier must be callable')
         super().__init__(*args)
 
-    def _fillcache(self, n: int) -> None:
+    def _fillcache(self, n: Optional[int]) -> None:
         """Cache `n` modified items. If `n` is 0 or None, 1 item is cached.
 
         Each item returned by the iterator is passed through the
diff --git a/sphinx/locale/__init__.py b/sphinx/locale/__init__.py
index 9812355ca..91adfef7c 100644
--- a/sphinx/locale/__init__.py
+++ b/sphinx/locale/__init__.py
@@ -12,7 +12,7 @@ import gettext
 import locale
 from collections import UserString, defaultdict
 from gettext import NullTranslations
-from typing import Any, Callable, Dict, Iterable, List, Tuple, Union
+from typing import Any, Callable, Dict, Iterable, List, Optional, Tuple, Union
 
 
 class _TranslationProxy(UserString):
@@ -173,7 +173,7 @@ def init_console(locale_dir: str, catalog: str) -> Tuple[NullTranslations, bool]
     """
     try:
         # encoding is ignored
-        language, _ = locale.getlocale(locale.LC_MESSAGES)
+        language, _ = locale.getlocale(locale.LC_MESSAGES)  # type: Tuple[Optional[str], Any]
     except AttributeError:
         # LC_MESSAGES is not always defined. Fallback to the default language
         # in case it is not.
diff --git a/sphinx/pycode/ast.py b/sphinx/pycode/ast.py
index c885db494..9664e7edb 100644
--- a/sphinx/pycode/ast.py
+++ b/sphinx/pycode/ast.py
@@ -58,7 +58,7 @@ def parse(code: str, mode: str = 'exec') -> "ast.AST":
         return ast.parse(code, mode=mode)
 
 
-def unparse(node: ast.AST) -> str:
+def unparse(node: Optional[ast.AST]) -> Optional[str]:
     """Unparse an AST to string."""
     if node is None:
         return None
@@ -138,7 +138,7 @@ def _unparse_arg(arg: ast.arg, default: Optional[ast.AST]) -> str:
 
 def unparse_arguments(node: ast.arguments) -> str:
     """Unparse an arguments to string."""
-    defaults = list(node.defaults)
+    defaults = list(node.defaults)  # type: List[Optional[ast.AST]]
     positionals = len(node.args)
     posonlyargs = 0
     if hasattr(node, "posonlyargs"):  # for py38+
@@ -147,7 +147,7 @@ def unparse_arguments(node: ast.arguments) -> str:
     for _ in range(len(defaults), positionals):
         defaults.insert(0, None)
 
-    kw_defaults = list(node.kw_defaults)
+    kw_defaults = list(node.kw_defaults)  # type: List[Optional[ast.AST]]
     for _ in range(len(kw_defaults), len(node.kwonlyargs)):
         kw_defaults.insert(0, None)
 
diff --git a/sphinx/util/images.py b/sphinx/util/images.py
index 568682b37..115007d31 100644
--- a/sphinx/util/images.py
+++ b/sphinx/util/images.py
@@ -12,7 +12,7 @@ import base64
 import imghdr
 from collections import OrderedDict
 from os import path
-from typing import IO, NamedTuple, Tuple
+from typing import IO, NamedTuple, Optional, Tuple
 
 import imagesize
 
@@ -36,7 +36,7 @@ DataURI = NamedTuple('DataURI', [('mimetype', str),
                                  ('data', bytes)])
 
 
-def get_image_size(filename: str) -> Tuple[int, int]:
+def get_image_size(filename: str) -> Optional[Tuple[int, int]]:
     try:
         size = imagesize.get(filename)
         if size[0] == -1:
@@ -53,7 +53,7 @@ def get_image_size(filename: str) -> Tuple[int, int]:
         return None
 
 
-def guess_mimetype_for_stream(stream: IO, default: str = None) -> str:
+def guess_mimetype_for_stream(stream: IO, default: Optional[str] = None) -> Optional[str]:
     imgtype = imghdr.what(stream)  # type: ignore
     if imgtype:
         return 'image/' + imgtype
@@ -61,7 +61,7 @@ def guess_mimetype_for_stream(stream: IO, default: str = None) -> str:
         return default
 
 
-def guess_mimetype(filename: str = '', default: str = None) -> str:
+def guess_mimetype(filename: str = '', default: Optional[str] = None) -> Optional[str]:
     _, ext = path.splitext(filename.lower())
     if ext in mime_suffixes:
         return mime_suffixes[ext]
@@ -72,7 +72,7 @@ def guess_mimetype(filename: str = '', default: str = None) -> str:
     return default
 
 
-def get_image_extension(mimetype: str) -> str:
+def get_image_extension(mimetype: str) -> Optional[str]:
     for ext, _mimetype in mime_suffixes.items():
         if mimetype == _mimetype:
             return ext
@@ -80,7 +80,7 @@ def get_image_extension(mimetype: str) -> str:
     return None
 
 
-def parse_data_uri(uri: str) -> DataURI:
+def parse_data_uri(uri: str) -> Optional[DataURI]:
     if not uri.startswith('data:'):
         return None
 
@@ -101,7 +101,7 @@ def parse_data_uri(uri: str) -> DataURI:
     return DataURI(mimetype, charset, image_data)
 
 
-def test_svg(h: bytes, f: IO) -> str:
+def test_svg(h: bytes, f: IO) -> Optional[str]:
     """An additional imghdr library helper; test the header is SVG's or not."""
     try:
         if '<svg' in h.decode().lower():
diff --git a/sphinx/util/inspect.py b/sphinx/util/inspect.py
index 64c1568f6..a93b327cb 100644
--- a/sphinx/util/inspect.py
+++ b/sphinx/util/inspect.py
@@ -20,7 +20,7 @@ from inspect import (  # NOQA
     Parameter, isclass, ismethod, ismethoddescriptor
 )
 from io import StringIO
-from typing import Any, Callable, Mapping, List, Tuple
+from typing import Any, Callable, Mapping, List, Optional, Tuple
 from typing import cast
 
 from sphinx.deprecation import RemovedInSphinx40Warning, RemovedInSphinx50Warning
@@ -565,7 +565,7 @@ class Signature:
         self.partialmethod_with_noargs = False
 
         try:
-            self.signature = inspect.signature(subject)
+            self.signature = inspect.signature(subject)  # type: Optional[inspect.Signature]
         except IndexError:
             # Until python 3.6.4, cpython has been crashed on inspection for
             # partialmethods not having any arguments.
diff --git a/sphinx/util/osutil.py b/sphinx/util/osutil.py
index af41df48b..248abd0e7 100644
--- a/sphinx/util/osutil.py
+++ b/sphinx/util/osutil.py
@@ -18,7 +18,7 @@ import sys
 import warnings
 from io import StringIO
 from os import path
-from typing import Any, Generator, Iterator, List, Tuple
+from typing import Any, Generator, Iterator, List, Optional, Tuple
 
 from sphinx.deprecation import RemovedInSphinx40Warning
 
@@ -206,7 +206,7 @@ class FileAvoidWrite:
     """
     def __init__(self, path: str) -> None:
         self._path = path
-        self._io = None  # type: StringIO
+        self._io = None  # type: Optional[StringIO]
 
     def write(self, data: str) -> None:
         if not self._io:
diff --git a/sphinx/util/png.py b/sphinx/util/png.py
index c7f73fd50..22c35d991 100644
--- a/sphinx/util/png.py
+++ b/sphinx/util/png.py
@@ -10,6 +10,7 @@
 
 import binascii
 import struct
+from typing import Optional
 
 
 LEN_IEND = 12
@@ -20,7 +21,7 @@ DEPTH_CHUNK_START = b'tEXtDepth\x00'
 IEND_CHUNK = b'\x00\x00\x00\x00IEND\xAE\x42\x60\x82'
 
 
-def read_png_depth(filename: str) -> int:
+def read_png_depth(filename: str) -> Optional[int]:
     """Read the special tEXt chunk indicating the depth from a PNG file."""
     with open(filename, 'rb') as f:
         f.seek(- (LEN_IEND + LEN_DEPTH), 2)