diff --git a/pyproject.toml b/pyproject.toml
index e0ca815d95..00c6333aa5 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -160,6 +160,7 @@ strict_concatenate = true
 
 check_untyped_defs = true
 disallow_untyped_decorators = true
+disallow_any_generics = true
 
 [[tool.mypy.overrides]]
 module = [
@@ -176,6 +177,21 @@ module = [
 ]
 check_untyped_defs = false
 
+[[tool.mypy.overrides]]
+module = [
+    "zarr.v3.abc.codec",
+    "zarr.v3.codecs.bytes",
+    "zarr.v3.codecs.pipeline",
+    "zarr.v3.codecs.sharding",
+    "zarr.v3.codecs.transpose",
+    "zarr.v3.array_v2",
+    "zarr.v3.array",
+    "zarr.v3.sync",
+    "zarr.convenience",
+    "zarr.meta",
+]
+disallow_any_generics = false
+
 
 [tool.pytest.ini_options]
 doctest_optionflags = [
diff --git a/src/zarr/_storage/store.py b/src/zarr/_storage/store.py
index 911af20fda..6e13b08cc7 100644
--- a/src/zarr/_storage/store.py
+++ b/src/zarr/_storage/store.py
@@ -14,7 +14,7 @@
 DEFAULT_ZARR_VERSION = 2
 
 
-class BaseStore(MutableMapping):
+class BaseStore(MutableMapping[str, Any]):
     """Abstract base class for store implementations.
 
     This is a thin wrapper over MutableMapping that provides methods to check
@@ -165,7 +165,7 @@ def rmdir(self, path: str = "") -> None:
 
 
 # allow MutableMapping for backwards compatibility
-StoreLike = Union[BaseStore, MutableMapping]
+StoreLike = Union[BaseStore, MutableMapping[str, Any]]
 
 
 def _path_to_prefix(path: Optional[str]) -> str:
diff --git a/src/zarr/attrs.py b/src/zarr/attrs.py
index 65f0423ecd..89cfefc22e 100644
--- a/src/zarr/attrs.py
+++ b/src/zarr/attrs.py
@@ -6,7 +6,7 @@
 from zarr.util import json_dumps
 
 
-class Attributes(MutableMapping):
+class Attributes(MutableMapping[str, Any]):
     """Class providing access to user attributes on an array or group. Should not be
     instantiated directly, will be available via the `.attrs` property of an array or
     group.
diff --git a/src/zarr/convenience.py b/src/zarr/convenience.py
index 615a019dc3..b357c26c55 100644
--- a/src/zarr/convenience.py
+++ b/src/zarr/convenience.py
@@ -20,9 +20,9 @@
 )
 from zarr.util import TreeViewer, buffer_size, normalize_storage_path
 
-from typing import Union
+from typing import Any, Union
 
-StoreLike = Union[BaseStore, MutableMapping, str, None]
+StoreLike = Union[BaseStore, MutableMapping[str, Any], str, None]
 
 _builtin_open = open  # builtin open is later shadowed by a local open function
 
diff --git a/src/zarr/hierarchy.py b/src/zarr/hierarchy.py
index e30d2d7996..9044c1681e 100644
--- a/src/zarr/hierarchy.py
+++ b/src/zarr/hierarchy.py
@@ -1,5 +1,6 @@
 from collections.abc import MutableMapping
 from itertools import islice
+from typing import Any
 
 import numpy as np
 
@@ -48,7 +49,7 @@
 )
 
 
-class Group(MutableMapping):
+class Group(MutableMapping[str, Any]):
     """Instantiate a group from an initialized store.
 
     Parameters
diff --git a/src/zarr/storage.py b/src/zarr/storage.py
index a7bd22a6b9..b98cee99dd 100644
--- a/src/zarr/storage.py
+++ b/src/zarr/storage.py
@@ -99,7 +99,7 @@
 
 Path = Union[str, bytes, None]
 # allow MutableMapping for backwards compatibility
-StoreLike = Union[BaseStore, MutableMapping]
+StoreLike = Union[BaseStore, MutableMapping[str, Any]]
 
 
 def contains_array(store: StoreLike, path: Path = None) -> bool:
@@ -202,7 +202,7 @@ def listdir(store: BaseStore, path: Path = None):
 
 def _getsize(store: BaseStore, path: Path = None) -> int:
     # compute from size of values
-    if path and path in store:
+    if isinstance(path, str) and path in store:
         v = store[path]
         size = buffer_size(v)
     else:
@@ -584,7 +584,7 @@ def _init_group_metadata(
         store[key] = encode_group_metadata(meta)
 
 
-def _dict_store_keys(d: Dict, prefix="", cls=dict):
+def _dict_store_keys(d: dict[str, Any], prefix="", cls=dict):
     for k in d.keys():
         v = d[k]
         if isinstance(v, cls):
diff --git a/src/zarr/util.py b/src/zarr/util.py
index 35ecc64bba..0588e1a558 100644
--- a/src/zarr/util.py
+++ b/src/zarr/util.py
@@ -20,6 +20,7 @@
 )
 
 import numpy as np
+import numpy.typing as npt
 from asciitree import BoxStyle, LeftAligned
 from asciitree.traversal import Traversal
 from numcodecs.compat import (
@@ -36,7 +37,7 @@
 ValueType = TypeVar("ValueType")
 
 
-def flatten(arg: Iterable) -> Iterable:
+def flatten(arg: Iterable[Any]) -> Iterable[Any]:
     for element in arg:
         if isinstance(element, Iterable) and not isinstance(element, (str, bytes)):
             yield from flatten(element)
@@ -179,7 +180,7 @@ def normalize_chunks(chunks: Any, shape: Tuple[int, ...], typesize: int) -> Tupl
     return chunks
 
 
-def normalize_dtype(dtype: Union[str, np.dtype], object_codec) -> Tuple[np.dtype, Any]:
+def normalize_dtype(dtype: Union[str, npt.DTypeLike], object_codec) -> Tuple[np.dtype[Any], Any]:
     # convenience API for object arrays
     if inspect.isclass(dtype):
         dtype = dtype.__name__
@@ -291,7 +292,7 @@ def normalize_dimension_separator(sep: Optional[str]) -> Optional[str]:
         raise ValueError("dimension_separator must be either '.' or '/', found: %r" % sep)
 
 
-def normalize_fill_value(fill_value, dtype: np.dtype):
+def normalize_fill_value(fill_value, dtype: np.dtype[Any]):
     if fill_value is None or dtype.hasobject:
         # no fill value
         pass
@@ -668,7 +669,7 @@ def read_full(self):
 
 
 def retry_call(
-    callabl: Callable,
+    callabl: Callable[..., Any],
     args=None,
     kwargs=None,
     exceptions: Tuple[Any, ...] = (),
diff --git a/src/zarr/v3/common.py b/src/zarr/v3/common.py
index 1caf83a764..9f0a5fdb63 100644
--- a/src/zarr/v3/common.py
+++ b/src/zarr/v3/common.py
@@ -1,5 +1,5 @@
 from __future__ import annotations
-from typing import TYPE_CHECKING, Union, Tuple, Iterable, Dict, List, TypeVar, overload
+from typing import TYPE_CHECKING, Union, Tuple, Iterable, Dict, List, TypeVar, overload, Any
 import asyncio
 import contextvars
 from dataclasses import dataclass
@@ -28,7 +28,7 @@ def product(tup: ChunkCoords) -> int:
     return functools.reduce(lambda x, y: x * y, tup, 1)
 
 
-T = TypeVar("T", bound=Tuple)
+T = TypeVar("T", bound=Tuple[Any, ...])
 V = TypeVar("V")
 
 
@@ -76,7 +76,7 @@ def parse_enum(data: JSON, cls: Type[E]) -> E:
 @dataclass(frozen=True)
 class ArraySpec:
     shape: ChunkCoords
-    dtype: np.dtype
+    dtype: np.dtype[Any]
     fill_value: Any
 
     def __init__(self, shape, dtype, fill_value):
@@ -102,7 +102,7 @@ def parse_name(data: JSON, expected: Optional[str] = None) -> str:
         raise TypeError(f"Expected a string, got an instance of {type(data)}.")
 
 
-def parse_configuration(data: JSON) -> dict:
+def parse_configuration(data: JSON) -> JSON:
     if not isinstance(data, dict):
         raise TypeError(f"Expected dict, got {type(data)}")
     return data
@@ -153,7 +153,7 @@ def parse_shapelike(data: Any) -> Tuple[int, ...]:
     return data_tuple
 
 
-def parse_dtype(data: Any) -> np.dtype:
+def parse_dtype(data: Any) -> np.dtype[Any]:
     # todo: real validation
     return np.dtype(data)
 
diff --git a/src/zarr/v3/metadata.py b/src/zarr/v3/metadata.py
index a5e8927311..573b8484f0 100644
--- a/src/zarr/v3/metadata.py
+++ b/src/zarr/v3/metadata.py
@@ -4,6 +4,7 @@
 from dataclasses import dataclass, field
 import json
 import numpy as np
+import numpy.typing as npt
 
 from zarr.v3.chunk_grids import ChunkGrid, RegularChunkGrid
 from zarr.v3.chunk_key_encodings import ChunkKeyEncoding, parse_separator
@@ -90,7 +91,7 @@ def to_numpy_shortname(self) -> str:
         return data_type_to_numpy[self]
 
     @classmethod
-    def from_dtype(cls, dtype: np.dtype) -> DataType:
+    def from_dtype(cls, dtype: np.dtype[Any]) -> DataType:
         dtype_to_data_type = {
             "|b1": "bool",
             "bool": "bool",
@@ -111,7 +112,7 @@ def from_dtype(cls, dtype: np.dtype) -> DataType:
 @dataclass(frozen=True)
 class ArrayMetadata(Metadata):
     shape: ChunkCoords
-    data_type: np.dtype
+    data_type: np.dtype[Any]
     chunk_grid: ChunkGrid
     chunk_key_encoding: ChunkKeyEncoding
     fill_value: Any
@@ -176,7 +177,7 @@ def _validate_metadata(self) -> None:
         self.codecs.validate(self)
 
     @property
-    def dtype(self) -> np.dtype:
+    def dtype(self) -> np.dtype[Any]:
         return self.data_type
 
     @property
@@ -238,7 +239,7 @@ def to_dict(self) -> Dict[str, Any]:
 class ArrayV2Metadata(Metadata):
     shape: ChunkCoords
     chunks: ChunkCoords
-    dtype: np.dtype
+    dtype: np.dtype[Any]
     fill_value: Union[None, int, float] = 0
     order: Literal["C", "F"] = "C"
     filters: Optional[List[Dict[str, Any]]] = None
@@ -251,7 +252,7 @@ def __init__(
         self,
         *,
         shape: ChunkCoords,
-        dtype: np.dtype,
+        dtype: npt.DTypeLike,
         chunks: ChunkCoords,
         fill_value: Any,
         order: Literal["C", "F"],