diff --git a/pymbolic/mapper/__init__.py b/pymbolic/mapper/__init__.py
index 22315f9..459f5a3 100644
--- a/pymbolic/mapper/__init__.py
+++ b/pymbolic/mapper/__init__.py
@@ -24,11 +24,20 @@
 """
 
 from abc import ABC, abstractmethod
-from typing import Any
+from typing import TYPE_CHECKING, Generic, Hashable, Iterable, TypeVar
+from warnings import warn
 
 from immutabledict import immutabledict
+from typing_extensions import ParamSpec, TypeAlias, TypeIs
 
 import pymbolic.primitives as primitives
+from pymbolic.typing import ExpressionT
+
+
+if TYPE_CHECKING:
+    import numpy as np
+
+    from pymbolic.rational import Rational
 
 
 __doc__ = """
@@ -96,14 +105,20 @@
 """
 
 
-try:
-    import numpy
+if TYPE_CHECKING:
+    import numpy as np
+
+    def is_numpy_array(val) -> TypeIs[np.ndarray]:
+        return isinstance(val, np.ndarray)
+else:
+    try:
+        import numpy as np
 
-    def is_numpy_array(val):
-        return isinstance(val, numpy.ndarray)
-except ImportError:
-    def is_numpy_array(ary):
-        return False
+        def is_numpy_array(val):
+            return isinstance(val, np.ndarray)
+    except ImportError:
+        def is_numpy_array(ary):
+            return False
 
 
 class UnsupportedExpressionError(ValueError):
@@ -112,7 +127,11 @@ class UnsupportedExpressionError(ValueError):
 
 # {{{ mapper base
 
-class Mapper:
+ResultT = TypeVar("ResultT")
+P = ParamSpec("P")
+
+
+class Mapper(Generic[ResultT, P]):
     """A visitor for trees of :class:`pymbolic.Expression`
     subclasses. Each expression-derived object is dispatched to the
     method named by the :attr:`pymbolic.Expression.mapper_method`
@@ -120,7 +139,8 @@ class Mapper:
     *mapper_method* in the method resolution order of the object.
     """
 
-    def handle_unsupported_expression(self, expr, *args, **kwargs):
+    def handle_unsupported_expression(self,
+            expr: object, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         """Mapper method that is invoked for
         :class:`pymbolic.Expression` subclasses for which a mapper
         method does not exist in this mapper.
@@ -130,7 +150,8 @@ def handle_unsupported_expression(self, expr, *args, **kwargs):
                 "{} cannot handle expressions of type {}".format(
                     type(self), type(expr)))
 
-    def __call__(self, expr, *args, **kwargs):
+    def __call__(self,
+             expr: ExpressionT, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         """Dispatch *expr* to its corresponding mapper method. Pass on
         ``*args`` and ``**kwargs`` unmodified.
 
@@ -162,7 +183,8 @@ def __call__(self, expr, *args, **kwargs):
 
     rec = __call__
 
-    def rec_fallback(self, expr, *args, **kwargs):
+    def rec_fallback(self,
+            expr: object, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         if isinstance(expr, primitives.Expression):
             for cls in type(expr).__mro__[1:]:
                 method_name = getattr(cls, "mapper_method", None)
@@ -175,56 +197,95 @@ def rec_fallback(self, expr, *args, **kwargs):
         else:
             return self.map_foreign(expr, *args, **kwargs)
 
-    def map_algebraic_leaf(self, expr, *args, **kwargs):
+    def map_algebraic_leaf(self,
+            expr: primitives.AlgebraicLeaf,
+            *args: P.args, **kwargs: P.kwargs) -> ResultT:
         raise NotImplementedError
 
-    def map_variable(self, expr, *args, **kwargs):
+    def map_variable(self,
+            expr: primitives.Variable, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.map_algebraic_leaf(expr, *args, **kwargs)
 
-    def map_subscript(self, expr, *args, **kwargs):
+    def map_subscript(self,
+            expr: primitives.Subscript, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.map_algebraic_leaf(expr, *args, **kwargs)
 
-    def map_call(self, expr, *args, **kwargs):
+    def map_call(self,
+            expr: primitives.Call, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.map_algebraic_leaf(expr, *args, **kwargs)
 
-    def map_lookup(self, expr, *args, **kwargs):
+    def map_call_with_kwargs(self,
+            expr: primitives.CallWithKwargs,
+            *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.map_algebraic_leaf(expr, *args, **kwargs)
 
-    def map_if_positive(self, expr, *args, **kwargs):
+    def map_lookup(self,
+            expr: primitives.Lookup, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.map_algebraic_leaf(expr, *args, **kwargs)
 
-    def map_rational(self, expr, *args, **kwargs):
-        return self.map_quotient(expr, *args, **kwargs)
+    def map_if(self,
+            expr: primitives.If, *args: P.args, **kwargs: P.kwargs) -> ResultT:
+        raise NotImplementedError
 
-    def map_quotient(self, expr, *args, **kwargs):
+    def map_rational(self,
+            expr: Rational, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         raise NotImplementedError
 
-    def map_constant(self, expr, *args, **kwargs):
+    def map_quotient(self,
+            expr: primitives.Quotient, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         raise NotImplementedError
 
-    def map_list(self, expr, *args, **kwargs):
+    def map_floor_div(self,
+            expr: primitives.FloorDiv, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         raise NotImplementedError
 
-    def map_tuple(self, expr, *args, **kwargs):
+    def map_remainder(self,
+            expr: primitives.Remainder, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         raise NotImplementedError
 
-    def map_numpy_array(self, expr, *args, **kwargs):
+    def map_constant(self,
+            expr: object, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         raise NotImplementedError
 
-    def map_nan(self, expr, *args, **kwargs):
+    def map_list(self,
+            expr: list[ExpressionT], *args: P.args, **kwargs: P.kwargs) -> ResultT:
+        raise NotImplementedError
+
+    def map_tuple(self,
+            expr: tuple[ExpressionT, ...],
+            *args: P.args, **kwargs: P.kwargs) -> ResultT:
+        raise NotImplementedError
+
+    def map_numpy_array(self,
+            expr: np.ndarray, *args: P.args, **kwargs: P.kwargs) -> ResultT:
+        raise NotImplementedError
+
+    def map_nan(self,
+                expr: primitives.NaN,
+                *args: P.args,
+                **kwargs: P.kwargs
+            ) -> ResultT:
         return self.map_algebraic_leaf(expr, *args, **kwargs)
 
-    def map_foreign(self, expr, *args, **kwargs):
+    def map_foreign(self,
+                expr: object,
+                *args: P.args,
+                **kwargs: P.kwargs
+            ) -> ResultT:
         """Mapper method dispatch for non-:mod:`pymbolic` objects."""
 
         if isinstance(expr, primitives.VALID_CONSTANT_CLASSES):
             return self.map_constant(expr, *args, **kwargs)
         elif is_numpy_array(expr):
             return self.map_numpy_array(expr, *args, **kwargs)
-        elif isinstance(expr, list):
-            return self.map_list(expr, *args, **kwargs)
         elif isinstance(expr, tuple):
             return self.map_tuple(expr, *args, **kwargs)
+        elif isinstance(expr, list):
+            warn("List found in expression graph. "
+                 "This is deprecated and will stop working in 2025. "
+                 "Use tuples instead.", DeprecationWarning, stacklevel=2
+             )
+            return self.map_list(expr, *args, **kwargs)
         else:
             raise ValueError(
                     "{} encountered invalid foreign object: {}".format(
@@ -234,17 +295,24 @@ def map_foreign(self, expr, *args, **kwargs):
 _NOT_IN_CACHE = object()
 
 
-class CachedMapper(Mapper):
+CacheKeyT: TypeAlias = Hashable
+
+
+class CachedMapper(Mapper[ResultT, P]):
     """
     A mapper that memoizes the mapped result for the expressions traversed.
 
     .. automethod:: get_cache_key
     """
-    def __init__(self):
-        self._cache: dict[Any, Any] = {}
+    def __init__(self) -> None:
+        self._cache: dict[CacheKeyT, ResultT] = {}
         Mapper.__init__(self)
 
-    def get_cache_key(self, expr, *args, **kwargs):
+    def get_cache_key(self,
+              expr: ExpressionT,
+              *args: P.args,
+              **kwargs: P.kwargs
+          ) -> CacheKeyT:
         """
         Returns the key corresponding to which the result of a mapper method is
         stored in the cache.
@@ -260,7 +328,11 @@ def get_cache_key(self, expr, *args, **kwargs):
         # and "4 == 4.0", but their traversal results cannot be re-used.
         return (type(expr), expr, args, immutabledict(kwargs))
 
-    def __call__(self, expr, *args, **kwargs):
+    def __call__(self,
+                 expr: ExpressionT,
+                 *args,
+                 **kwargs
+             ) -> ResultT:
         result = self._cache.get(
                 (cache_key := self.get_cache_key(expr, *args, **kwargs)),
                 _NOT_IN_CACHE)
@@ -286,7 +358,10 @@ def __call__(self, expr, *args, **kwargs):
 
 # {{{ combine mapper
 
-class CombineMapper(Mapper):
+CombineArgT = TypeVar("CombineArgT")
+
+
+class CombineMapper(Mapper[ResultT, P], Generic[ResultT, P, CombineArgT]):
     """A mapper whose goal it is to *combine* all branches of the expression
     tree into one final result. The default implementation of all mapper
     methods simply recurse (:meth:`Mapper.rec`) on all branches emanating from
@@ -304,16 +379,19 @@ class CombineMapper(Mapper):
     :class:`pymbolic.mapper.dependency.DependencyMapper` is another example.
     """
 
-    def combine(self, values):
+    def combine(self, values: Iterable[CombineArgT]) -> ResultT:
         raise NotImplementedError
 
-    def map_call(self, expr, *args, **kwargs):
+    def map_call(self,
+            expr: primitives.Call, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.combine((
             self.rec(expr.function, *args, **kwargs),
             *[self.rec(child, *args, **kwargs) for child in expr.parameters]
             ))
 
-    def map_call_with_kwargs(self, expr, *args, **kwargs):
+    def map_call_with_kwargs(self,
+            expr: primitives.CallWithKwargs,
+            *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.combine((
             self.rec(expr.function, *args, **kwargs),
             *[self.rec(child, *args, **kwargs) for child in expr.parameters],
@@ -321,38 +399,49 @@ def map_call_with_kwargs(self, expr, *args, **kwargs):
               for child in expr.kw_parameters.values()]
             ))
 
-    def map_subscript(self, expr, *args, **kwargs):
+    def map_subscript(self,
+            expr: primitives.Subscript, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.combine(
                 [self.rec(expr.aggregate, *args, **kwargs),
                     self.rec(expr.index, *args, **kwargs)])
 
-    def map_lookup(self, expr, *args, **kwargs):
+    def map_lookup(self,
+            expr: primitives.Lookup, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.rec(expr.aggregate, *args, **kwargs)
 
-    def map_sum(self, expr, *args, **kwargs):
+    def map_sum(self,
+            expr: primitives.Sum, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.combine(self.rec(child, *args, **kwargs)
                 for child in expr.children)
 
-    map_product = map_sum
+    def map_product(self,
+            expr: primitives.Product, *args: P.args, **kwargs: P.kwargs) -> ResultT:
+        return self.combine(self.rec(child, *args, **kwargs)
+                for child in expr.children)
 
-    def map_quotient(self, expr, *args, **kwargs):
+    def map_quotient(self,
+            expr: primitives.Quotient, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.combine((
             self.rec(expr.numerator, *args, **kwargs),
             self.rec(expr.denominator, *args, **kwargs)))
 
-    map_floor_div = map_quotient
-    map_remainder = map_quotient
+    def map_floor_div(self,
+            expr: primitives.FloorDiv, *args: P.args, **kwargs: P.kwargs) -> ResultT:
+        return self.combine((
+            self.rec(expr.numerator, *args, **kwargs),
+            self.rec(expr.denominator, *args, **kwargs)))
 
-    def map_power(self, expr, *args, **kwargs):
+    def map_remainder(self,
+            expr: primitives.Remainder, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.combine((
-                self.rec(expr.base, *args, **kwargs),
-                self.rec(expr.exponent, *args, **kwargs)))
+            self.rec(expr.numerator, *args, **kwargs),
+            self.rec(expr.denominator, *args, **kwargs)))
 
-    def map_polynomial(self, expr, *args, **kwargs):
+    def map_power(self,
+            expr: primitives.Power, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         return self.combine((
-            self.rec(expr.base, *args, **kwargs),
-            *[self.rec(coeff, *args, **kwargs) for exp, coeff in expr.data]
-            ))
+                self.rec(expr.base, *args, **kwargs),
+                self.rec(expr.exponent, *args, **kwargs)))
 
     def map_left_shift(self, expr, *args, **kwargs):
         return self.combine((