diff --git a/numpy/__init__.pyi b/numpy/__init__.pyi
index bf91aafdd3ed..d275ad27a45e 100644
--- a/numpy/__init__.pyi
+++ b/numpy/__init__.pyi
@@ -862,6 +862,24 @@ class dtype(Generic[_DTypeScalar]):
         align: bool = ...,
         copy: bool = ...,
     ) -> dtype[void]: ...
+
+    @overload
+    def __getitem__(self: dtype[void], key: List[str]) -> dtype[void]: ...
+    @overload
+    def __getitem__(self: dtype[void], key: Union[str, int]) -> dtype[Any]: ...
+
+    # NOTE: In the future 1-based multiplications will also yield `void` dtypes
+    @overload
+    def __mul__(self, value: Literal[0]) -> None: ...  # type: ignore[misc]
+    @overload
+    def __mul__(self, value: Literal[1]) -> dtype[_DTypeScalar]: ...
+    @overload
+    def __mul__(self, value: int) -> dtype[void]: ...
+
+    # NOTE: `__rmul__` seems to be broken when used in combination with
+    # literals as of mypy 0.800. Set the return-type to `Any` for now.
+    def __rmul__(self, value: int) -> Any: ...
+
     def __eq__(self, other: DTypeLike) -> bool: ...
     def __ne__(self, other: DTypeLike) -> bool: ...
     def __gt__(self, other: DTypeLike) -> bool: ...
@@ -901,6 +919,8 @@ class dtype(Generic[_DTypeScalar]):
     @property
     def name(self) -> str: ...
     @property
+    def names(self) -> Optional[Tuple[str, ...]]: ...
+    @property
     def num(self) -> int: ...
     @property
     def shape(self) -> _Shape: ...
diff --git a/numpy/typing/tests/data/pass/dtype.py b/numpy/typing/tests/data/pass/dtype.py
index cbae8c078551..90715209eac2 100644
--- a/numpy/typing/tests/data/pass/dtype.py
+++ b/numpy/typing/tests/data/pass/dtype.py
@@ -1,5 +1,8 @@
 import numpy as np
 
+dtype_obj = np.dtype(np.str_)
+void_dtype_obj = np.dtype([("f0", np.float64), ("f1", np.float32)])
+
 np.dtype(dtype=np.int64)
 np.dtype(int)
 np.dtype("int")
@@ -33,3 +36,16 @@ class Test:
 
 
 np.dtype(Test())
+
+dtype_obj.names
+
+dtype_obj * 0
+dtype_obj * 2
+
+0 * dtype_obj
+2 * dtype_obj
+
+void_dtype_obj["f0"]
+void_dtype_obj[0]
+void_dtype_obj[["f0", "f1"]]
+void_dtype_obj[["f0"]]
diff --git a/numpy/typing/tests/data/reveal/dtype.py b/numpy/typing/tests/data/reveal/dtype.py
index d414f2c4934f..83a7695b9bb5 100644
--- a/numpy/typing/tests/data/reveal/dtype.py
+++ b/numpy/typing/tests/data/reveal/dtype.py
@@ -1,5 +1,8 @@
 import numpy as np
 
+dtype_obj: np.dtype[np.str_]
+void_dtype_obj: np.dtype[np.void]
+
 reveal_type(np.dtype(np.float64))  # E: numpy.dtype[numpy.floating[numpy.typing._64Bit]]
 reveal_type(np.dtype(np.int64))  # E: numpy.dtype[numpy.signedinteger[numpy.typing._64Bit]]
 
@@ -31,3 +34,19 @@
 
 # Void
 reveal_type(np.dtype(("U", 10)))  # E: numpy.dtype[numpy.void]
+
+reveal_type(dtype_obj.name)  # E: str
+reveal_type(dtype_obj.names)  # E: Union[builtins.tuple[builtins.str], None]
+
+reveal_type(dtype_obj * 0)  # E: None
+reveal_type(dtype_obj * 1)  # E: numpy.dtype[numpy.str_]
+reveal_type(dtype_obj * 2)  # E: numpy.dtype[numpy.void]
+
+reveal_type(0 * dtype_obj)  # E: Any
+reveal_type(1 * dtype_obj)  # E: Any
+reveal_type(2 * dtype_obj)  # E: Any
+
+reveal_type(void_dtype_obj["f0"])  # E: numpy.dtype[Any]
+reveal_type(void_dtype_obj[0])  # E: numpy.dtype[Any]
+reveal_type(void_dtype_obj[["f0", "f1"]])  # E: numpy.dtype[numpy.void]
+reveal_type(void_dtype_obj[["f0"]])  # E: numpy.dtype[numpy.void]