diff --git a/ci/environment-typecheck.yml b/ci/environment-typecheck.yml
index 1b7c482f5..688fe9cc3 100644
--- a/ci/environment-typecheck.yml
+++ b/ci/environment-typecheck.yml
@@ -2,7 +2,7 @@ name: test_env
 channels:
   - conda-forge
 dependencies:
-  - mypy=1.4.1
+  - mypy=1.6.1
   - pyarrow
   - python=3.8
   - pip
diff --git a/fsspec/dircache.py b/fsspec/dircache.py
index eca19566b..33ecf44b7 100644
--- a/fsspec/dircache.py
+++ b/fsspec/dircache.py
@@ -1,9 +1,15 @@
+from __future__ import annotations
+
 import time
-from collections.abc import MutableMapping
 from functools import lru_cache
+from typing import Any, Iterator, MutableMapping, TypedDict, TypeAlias, TYPE_CHECKING
+
+if TYPE_CHECKING:
+    # TODO: consider a more-precise type using TypedDict
+    DirEntry: TypeAlias = dict[str, Any]
 
 
-class DirCache(MutableMapping):
+class DirCache(MutableMapping[str, list[DirEntry]]):
     """
     Caching of directory listings, in a structure like::
 
@@ -26,10 +32,10 @@ class DirCache(MutableMapping):
 
     def __init__(
         self,
-        use_listings_cache=True,
-        listings_expiry_time=None,
-        max_paths=None,
-        **kwargs,
+        use_listings_cache: bool = True,
+        listings_expiry_time: float | None = None,
+        max_paths: int | None = None,
+        **kwargs: Any,
     ):
         """
 
@@ -45,15 +51,15 @@ def __init__(
             The number of most recent listings that are considered valid; 'recent'
             refers to when the entry was set.
         """
-        self._cache = {}
-        self._times = {}
+        self._cache: dict[str, list[DirEntry]] = {}
+        self._times: dict[str, float] = {}
         if max_paths:
             self._q = lru_cache(max_paths + 1)(lambda key: self._cache.pop(key, None))
         self.use_listings_cache = use_listings_cache
         self.listings_expiry_time = listings_expiry_time
         self.max_paths = max_paths
 
-    def __getitem__(self, item):
+    def __getitem__(self, item: str) -> list[DirEntry]:
         if self.listings_expiry_time is not None:
             if self._times.get(item, 0) - time.time() < -self.listings_expiry_time:
                 del self._cache[item]
@@ -61,20 +67,20 @@ def __getitem__(self, item):
             self._q(item)
         return self._cache[item]  # maybe raises KeyError
 
-    def clear(self):
+    def clear(self) -> None:
         self._cache.clear()
 
-    def __len__(self):
+    def __len__(self) -> int:
         return len(self._cache)
 
-    def __contains__(self, item):
+    def __contains__(self, item: object) -> bool:
         try:
-            self[item]
+            self[item]  # type: ignore[index]
             return True
         except KeyError:
             return False
 
-    def __setitem__(self, key, value):
+    def __setitem__(self, key: str, value: List[DirEntry]) -> None:
         if not self.use_listings_cache:
             return
         if self.max_paths:
@@ -83,10 +89,10 @@ def __setitem__(self, key, value):
         if self.listings_expiry_time is not None:
             self._times[key] = time.time()
 
-    def __delitem__(self, key):
+    def __delitem__(self, key: str) -> None:
         del self._cache[key]
 
-    def __iter__(self):
+    def __iter__(self) -> Iterator[str]:
         entries = list(self._cache)
 
         return (k for k in entries if k in self)
diff --git a/fsspec/mapping.py b/fsspec/mapping.py
index b9822ae17..a8bbaa7e4 100644
--- a/fsspec/mapping.py
+++ b/fsspec/mapping.py
@@ -1,13 +1,29 @@
+from __future__ import annotations
+
 import array
 import posixpath
 import warnings
-from collections.abc import MutableMapping
 from functools import cached_property
+from typing import (
+    TYPE_CHECKING,
+    Iterable,
+    Iterator,
+    Literal,
+    MutableMapping,
+    TypeVar,
+    overload,
+)
 
 from .core import url_to_fs
 
+if TYPE_CHECKING:
+    from .spec import AbstractFileSystem
+    from .implementations.dirfs import DirFileSystem
+
+    T = TypeVar("T")
+
 
-class FSMap(MutableMapping):
+class FSMap(MutableMapping[str, bytes]):
     """Wrap a FileSystem instance as a mutable wrapping.
 
     The keys of the mapping become files under the given root, and the
@@ -35,7 +51,14 @@ class FSMap(MutableMapping):
     b'Hello World'
     """
 
-    def __init__(self, root, fs, check=False, create=False, missing_exceptions=None):
+    def __init__(
+        self,
+        root: str,
+        fs: AbstractFileSystem,
+        check: bool = False,
+        create: bool = False,
+        missing_exceptions: tuple[type[Exception], ...] | None = None,
+    ):
         self.fs = fs
         self.root = fs._strip_protocol(root).rstrip("/")
         self._root_key_to_str = fs._strip_protocol(posixpath.join(root, "x"))[:-1]
@@ -61,13 +84,13 @@ def __init__(self, root, fs, check=False, create=False, missing_exceptions=None)
             self.fs.rm(root + "/a")
 
     @cached_property
-    def dirfs(self):
+    def dirfs(self) -> DirFileSystem:
         """dirfs instance that can be used with the same keys as the mapper"""
         from .implementations.dirfs import DirFileSystem
 
         return DirFileSystem(path=self._root_key_to_str, fs=self.fs)
 
-    def clear(self):
+    def clear(self) -> None:
         """Remove all keys below root - empties out mapping"""
         try:
             self.fs.rm(self.root, True)
@@ -75,14 +98,30 @@ def clear(self):
         except:  # noqa: E722
             pass
 
-    def getitems(self, keys, on_error="raise"):
+    @overload
+    def getitems(
+        self, keys: Iterable[str], on_error: Literal["raise", "omit"] = ...
+    ) -> dict[str, bytes]:
+        pass
+
+    @overload
+    def getitems(
+        self, keys: Iterable[str], on_error: Literal["return"]
+    ) -> dict[str, bytes | Exception]:
+        pass
+
+    def getitems(
+        self,
+        keys: Iterable[str],
+        on_error: Literal["raise", "omit", "return"] = "raise",
+    ) -> dict[str, bytes | Exception] | dict[str, bytes]:
         """Fetch multiple items from the store
 
         If the backend is async-able, this might proceed concurrently
 
         Parameters
         ----------
-        keys: list(str)
+        keys: iterable(str)
             They keys to be fetched
         on_error : "raise", "omit", "return"
             If raise, an underlying exception will be raised (converted to KeyError
@@ -113,7 +152,7 @@ def getitems(self, keys, on_error="raise"):
             if on_error == "return" or not isinstance(out[k2], BaseException)
         }
 
-    def setitems(self, values_dict):
+    def setitems(self, values_dict: dict[str, bytes]) -> None:
         """Set the values of multiple items in the store
 
         Parameters
@@ -123,11 +162,11 @@ def setitems(self, values_dict):
         values = {self._key_to_str(k): maybe_convert(v) for k, v in values_dict.items()}
         self.fs.pipe(values)
 
-    def delitems(self, keys):
+    def delitems(self, keys: Iterable[str]) -> None:
         """Remove multiple keys from the store"""
         self.fs.rm([self._key_to_str(k) for k in keys])
 
-    def _key_to_str(self, key):
+    def _key_to_str(self, key: str) -> str:
         """Generate full path for the key"""
         if not isinstance(key, str):
             # raise TypeError("key must be of type `str`, got `{type(key).__name__}`"
@@ -140,11 +179,11 @@ def _key_to_str(self, key):
             key = str(key)
         return f"{self._root_key_to_str}{key}"
 
-    def _str_to_key(self, s):
+    def _str_to_key(self, s: str) -> str:
         """Strip path of to leave key name"""
         return s[len(self.root) :].lstrip("/")
 
-    def __getitem__(self, key, default=None):
+    def __getitem__(self, key: str, default: bytes | None = None) -> bytes:
         """Retrieve data"""
         k = self._key_to_str(key)
         try:
@@ -155,7 +194,7 @@ def __getitem__(self, key, default=None):
             raise KeyError(key)
         return result
 
-    def pop(self, key, default=None):
+    def pop(self, key: str, default: bytes | None = None) -> bytes:  # type: ignore[override]
         """Pop data"""
         result = self.__getitem__(key, default)
         try:
@@ -164,26 +203,26 @@ def pop(self, key, default=None):
             pass
         return result
 
-    def __setitem__(self, key, value):
+    def __setitem__(self, key: str, value: bytes) -> None:
         """Store value in key"""
         key = self._key_to_str(key)
         self.fs.mkdirs(self.fs._parent(key), exist_ok=True)
         self.fs.pipe_file(key, maybe_convert(value))
 
-    def __iter__(self):
+    def __iter__(self) -> Iterator[str]:
         return (self._str_to_key(x) for x in self.fs.find(self.root))
 
-    def __len__(self):
+    def __len__(self) -> int:
         return len(self.fs.find(self.root))
 
-    def __delitem__(self, key):
+    def __delitem__(self, key: str) -> None:
         """Remove key"""
         try:
             self.fs.rm(self._key_to_str(key))
         except:  # noqa: E722
             raise KeyError
 
-    def __contains__(self, key):
+    def __contains__(self, key: str) -> bool:  # type: ignore[override]
         """Does key exist in mapping?"""
         path = self._key_to_str(key)
         return self.fs.exists(path) and self.fs.isfile(path)
@@ -204,13 +243,13 @@ def maybe_convert(value):
 
 
 def get_mapper(
-    url="",
-    check=False,
-    create=False,
-    missing_exceptions=None,
-    alternate_root=None,
+    url: str = "",
+    check: bool = False,
+    create: bool = False,
+    missing_exceptions: tuple[type[Exception], ...] | None = None,
+    alternate_root: str | None = None,
     **kwargs,
-):
+) -> FSMap:
     """Create key-value interface for given URL and options
 
     The URL will be of the form "protocol://location" and point to the root
diff --git a/setup.cfg b/setup.cfg
index 67467cd20..a8c2e7503 100644
--- a/setup.cfg
+++ b/setup.cfg
@@ -57,5 +57,11 @@ exclude = (test.*|conftest)\.py$
 [mypy-fsspec.caching]
 check_untyped_defs = True
 
+[mypy-fsspec.dircache]
+check_untyped_defs = True
+
+[mypy-fsspec.mapping]
+check_untyped_defs = True
+
 [mypy-fsspec.utils]
 check_untyped_defs = True