diff --git a/jax/_src/api.py b/jax/_src/api.py
index bd8a951954ac..053ff26d933e 100644
--- a/jax/_src/api.py
+++ b/jax/_src/api.py
@@ -139,7 +139,7 @@ def _update_debug_special_thread_local(_):
 
 
 def jit(
-  fun: Callable,
+  fun: Callable[stages._P, stages._OutT],
   in_shardings=sharding_impls.UNSPECIFIED,
   out_shardings=sharding_impls.UNSPECIFIED,
   static_argnums: int | Sequence[int] | None = None,
@@ -151,7 +151,7 @@ def jit(
   backend: str | None = None,
   inline: bool = False,
   abstracted_axes: Any | None = None,
-) -> pjit.JitWrapped:
+) -> pjit.JitWrapped[stages._P, stages._OutT]:
   """Sets up ``fun`` for just-in-time compilation with XLA.
 
   Args:
diff --git a/jax/_src/lax/linalg.py b/jax/_src/lax/linalg.py
index ef6a5a11a56e..bbef2c870c7c 100644
--- a/jax/_src/lax/linalg.py
+++ b/jax/_src/lax/linalg.py
@@ -1649,7 +1649,7 @@ def _lu_solve(lu: Array, permutation: Array, b: Array, trans: int) -> Array:
 def lu_solve(lu: ArrayLike, permutation: ArrayLike, b: ArrayLike,
              trans: int = 0) -> Array:
   """LU solve with broadcasting."""
-  return _lu_solve(lu, permutation, b, trans)
+  return _lu_solve(lu, permutation, b, trans)  # type: ignore[arg-type]
 
 
 # QR decomposition
diff --git a/jax/_src/numpy/array_methods.py b/jax/_src/numpy/array_methods.py
index 547fe1247459..bb198bdbc677 100644
--- a/jax/_src/numpy/array_methods.py
+++ b/jax/_src/numpy/array_methods.py
@@ -168,7 +168,7 @@ def _cumprod(self: Array, axis: reductions.Axis = None, dtype: DTypeLike | None
 
   Refer to :func:`jax.numpy.cumprod` for the full documentation.
   """
-  return reductions.cumprod(self, axis=axis, dtype=dtype, out=out)
+  return reductions.cumprod(self, axis=axis, dtype=dtype, out=out)  # type: ignore[arg-type]
 
 def _cumsum(self: Array, axis: reductions.Axis = None, dtype: DTypeLike | None = None,
             out: None = None) -> Array:
@@ -176,7 +176,7 @@ def _cumsum(self: Array, axis: reductions.Axis = None, dtype: DTypeLike | None =
 
   Refer to :func:`jax.numpy.cumsum` for the full documentation.
   """
-  return reductions.cumsum(self, axis=axis, dtype=dtype, out=out)
+  return reductions.cumsum(self, axis=axis, dtype=dtype, out=out)  # type: ignore[arg-type]
 
 def _diagonal(self: Array, offset: int = 0, axis1: int = 0, axis2: int = 1) -> Array:
   """Return the specified diagonal from the array.
diff --git a/jax/_src/numpy/lax_numpy.py b/jax/_src/numpy/lax_numpy.py
index b71412e586bd..6ba4e41245f6 100644
--- a/jax/_src/numpy/lax_numpy.py
+++ b/jax/_src/numpy/lax_numpy.py
@@ -1124,7 +1124,7 @@ def flip(m: ArrayLike, axis: int | Sequence[int] | None = None) -> Array:
             [6, 5]]], dtype=int32)
   """
   util.check_arraylike("flip", m)
-  return _flip(asarray(m), reductions._ensure_optional_axes(axis))
+  return _flip(asarray(m), reductions._ensure_optional_axes(axis))  # type: ignore[arg-type]
 
 @partial(jit, static_argnames=('axis',))
 def _flip(m: Array, axis: int | tuple[int, ...] | None = None) -> Array:
@@ -1982,7 +1982,7 @@ def squeeze(a: ArrayLike, axis: int | Sequence[int] | None = None) -> Array:
     Array([0, 1, 2], dtype=int32)
   """
   util.check_arraylike("squeeze", a)
-  return _squeeze(asarray(a), _ensure_index_tuple(axis) if axis is not None else None)
+  return _squeeze(asarray(a), _ensure_index_tuple(axis) if axis is not None else None)  # type: ignore[arg-type]
 
 @partial(jit, static_argnames=('axis',), inline=True)
 def _squeeze(a: Array, axis: tuple[int, ...]) -> Array:
@@ -7259,7 +7259,7 @@ def delete(
     obj = asarray(obj).ravel()
     obj = clip(where(obj < 0, obj + a.shape[axis], obj), 0, a.shape[axis])
     obj = sort(obj)
-    obj -= arange(len(obj))  # type: ignore[arg-type,operator]
+    obj -= arange(len(obj))
     i = arange(a.shape[axis] - obj.size)
     i += (i[None, :] >= obj[:, None]).sum(0)
     return a[(slice(None),) * axis + (i,)]
@@ -11034,8 +11034,8 @@ def digitize(x: ArrayLike, bins: ArrayLike, right: bool = False,
   kwds: dict[str, str] = {} if method is None else {'method': method}
   return where(
     bins_arr[-1] >= bins_arr[0],
-    searchsorted(bins_arr, x, side=side, **kwds),
-    bins_arr.shape[0] - searchsorted(bins_arr[::-1], x, side=side, **kwds)
+    searchsorted(bins_arr, x, side=side, **kwds),  # type: ignore[arg-type]
+    bins_arr.shape[0] - searchsorted(bins_arr[::-1], x, side=side, **kwds)  # type: ignore[arg-type]
   )
 
 
diff --git a/jax/_src/numpy/ufunc_api.py b/jax/_src/numpy/ufunc_api.py
index 3473e8a7468a..96ced5abb45d 100644
--- a/jax/_src/numpy/ufunc_api.py
+++ b/jax/_src/numpy/ufunc_api.py
@@ -174,7 +174,7 @@ def __call__(self, *args: ArrayLike, out: None = None, where: None = None) -> An
     if where is not None:
       raise NotImplementedError(f"where argument of {self}")
     call = self.__static_props['call'] or self._call_vectorized
-    return call(*args)
+    return call(*args)  # type: ignore[arg-type]
 
   @partial(jax.jit, static_argnames=['self'])
   def _call_vectorized(self, *args):
diff --git a/jax/_src/pjit.py b/jax/_src/pjit.py
index ac1318ed7810..f940dce14139 100644
--- a/jax/_src/pjit.py
+++ b/jax/_src/pjit.py
@@ -808,7 +808,7 @@ def ax_leaf(l):
   return broadcast_prefix(abstracted_axes, args, ax_leaf)
 
 
-class JitWrapped(stages.Wrapped):
+class JitWrapped(stages.Wrapped[stages._P, stages._OutT]):
 
   def eval_shape(self, *args, **kwargs):
     """See ``jax.eval_shape``."""
diff --git a/jax/_src/scipy/cluster/vq.py b/jax/_src/scipy/cluster/vq.py
index a82c8928644d..05cad3504b98 100644
--- a/jax/_src/scipy/cluster/vq.py
+++ b/jax/_src/scipy/cluster/vq.py
@@ -70,5 +70,5 @@ def vq(obs: ArrayLike, code_book: ArrayLike, check_finite: bool = True) -> tuple
       raise ValueError("ndim different than 1 or 2 are not supported")
   dist = vmap(lambda ob: jnp.linalg.norm(ob[None] - cb_arr, axis=-1))(obs_arr)
   code = jnp.argmin(dist, axis=-1)
-  dist_min = vmap(operator.getitem)(dist, code)
+  dist_min = vmap(operator.getitem)(dist, code)  # type: ignore[call-overload]
   return code, dist_min
diff --git a/jax/_src/scipy/linalg.py b/jax/_src/scipy/linalg.py
index d014e5ceb24e..9066076c5fbf 100644
--- a/jax/_src/scipy/linalg.py
+++ b/jax/_src/scipy/linalg.py
@@ -219,7 +219,7 @@ def _svd(x: ArrayLike, *, full_matrices: bool, compute_uv: Literal[False]) -> Ar
 @overload
 def _svd(x: ArrayLike, *, full_matrices: bool, compute_uv: bool) -> Array | tuple[Array, Array, Array]: ...
 
-@partial(jit, static_argnames=('full_matrices', 'compute_uv'))
+@partial(jit, static_argnames=('full_matrices', 'compute_uv'))  # type: ignore[misc]
 def _svd(a: ArrayLike, *, full_matrices: bool, compute_uv: bool) -> Array | tuple[Array, Array, Array]:
   a, = promote_dtypes_inexact(jnp.asarray(a))
   return lax_linalg.svd(a, full_matrices=full_matrices, compute_uv=compute_uv)
@@ -545,7 +545,7 @@ def schur(a: ArrayLike, output: str = 'real') -> tuple[Array, Array]:
   if output not in ('real', 'complex'):
     raise ValueError(
       f"Expected 'output' to be either 'real' or 'complex', got {output=}.")
-  return _schur(a, output)
+  return _schur(a, output)  # type: ignore[arg-type]
 
 
 def inv(a: ArrayLike, overwrite_a: bool = False, check_finite: bool = True) -> Array:
diff --git a/jax/_src/scipy/special.py b/jax/_src/scipy/special.py
index 837aa011f165..77d7db65c737 100644
--- a/jax/_src/scipy/special.py
+++ b/jax/_src/scipy/special.py
@@ -1805,7 +1805,7 @@ def sph_harm(m: Array,
       int, n_max, 'The `n_max` argument of `jnp.scipy.special.sph_harm` must '
       'be statically specified to use `sph_harm` within JAX transformations.')
 
-  return _sph_harm(m, n, theta, phi, n_max)
+  return _sph_harm(m, n, theta, phi, n_max)  # type: ignore[arg-type]
 
 
 # exponential integrals
diff --git a/jax/_src/stages.py b/jax/_src/stages.py
index 3a2c375b64db..41d5cdc6c785 100644
--- a/jax/_src/stages.py
+++ b/jax/_src/stages.py
@@ -33,7 +33,7 @@
 import functools
 from collections.abc import Sequence
 from dataclasses import dataclass
-from typing import Any, NamedTuple, Protocol, Union, runtime_checkable
+from typing import Any, NamedTuple, ParamSpec, Protocol, TypeVar, Union, runtime_checkable
 
 import jax
 
@@ -751,9 +751,11 @@ def lower(self, *, lowering_platforms: tuple[str, ...] | None = None,
       raise ValueError(msg) from None
     return Lowered(lowering, self.args_info, self._out_tree)
 
+_P = ParamSpec("_P")
+_OutT = TypeVar("_OutT", covariant=True)  # pytype: disable=not-supported-yet
 
 @runtime_checkable
-class Wrapped(Protocol):
+class Wrapped(Protocol[_P, _OutT]):
   """A function ready to be traced, lowered, and compiled.
 
   This protocol reflects the output of functions such as
@@ -762,11 +764,11 @@ class Wrapped(Protocol):
   to compilation, and the result compiled prior to execution.
   """
 
-  def __call__(self, *args, **kwargs):
+  def __call__(self, *args: _P.args, **kwargs: _P.kwargs) -> _OutT:
     """Executes the wrapped function, lowering and compiling as needed."""
     raise NotImplementedError
 
-  def trace(self, *args, **kwargs) -> Traced:
+  def trace(self, *args: _P.args, **kwargs: _P.kwargs) -> Traced:
     """Trace this function explicitly for the given arguments.
 
     A traced function is staged out of Python and translated to a jaxpr. It is
@@ -777,7 +779,7 @@ def trace(self, *args, **kwargs) -> Traced:
     """
     raise NotImplementedError
 
-  def lower(self, *args, **kwargs) -> Lowered:
+  def lower(self, *args: _P.args, **kwargs: _P.kwargs) -> Lowered:
     """Lower this function explicitly for the given arguments.
 
     A lowered function is staged out of Python and translated to a
diff --git a/jax/_src/third_party/scipy/special.py b/jax/_src/third_party/scipy/special.py
index 67ef09f6de37..3c565f04623f 100644
--- a/jax/_src/third_party/scipy/special.py
+++ b/jax/_src/third_party/scipy/special.py
@@ -272,8 +272,8 @@ def fresnel(x: ArrayLike) -> tuple[Array, Array]:
     c_large = c_inf
     s_large = s_inf
   else:
-    c_large = 0.5 + 1 / (jnp.pi * x) * sinpi
-    s_large = 0.5 - 1 / (jnp.pi * x) * cospi
+    c_large = 0.5 + 1 / (jnp.pi * x) * sinpi  # type: ignore[assignment]
+    s_large = 0.5 - 1 / (jnp.pi * x) * cospi  # type: ignore[assignment]
 
   # Other x values
   t = jnp.pi * x2