diff --git a/src/pystencils/nbackend/kernelcreation/domain_kernels.py b/src/pystencils/nbackend/kernelcreation/domain_kernels.py
index a66cbd12ef05e57ffc972910045aff964d7fb2a1..1ee1cfae95463b39261b75a0dc6557e9431fe203 100644
--- a/src/pystencils/nbackend/kernelcreation/domain_kernels.py
+++ b/src/pystencils/nbackend/kernelcreation/domain_kernels.py
@@ -9,8 +9,11 @@ from ..ast import PsBlock
 
 from .context import KernelCreationContext, FullIterationSpace
 from .freeze import FreezeExpressions
+from .typification import Typifier
 
 # flake8: noqa
+
+
 def create_domain_kernel(assignments: AssignmentCollection):
     #   TODO: Assemble configuration
 
@@ -46,7 +49,8 @@ def create_domain_kernel(assignments: AssignmentCollection):
 
     #   5. Typify
     #   Also the same for both types of kernels
-    #   determine_types(kernel_body)
+    typify = Typifier(ctx)
+    kernel_body = typify(kernel_body)
 
     #   Up to this point, all was target-agnostic, but now the target becomes relevant.
     #   Here we might hand off the compilation to a target-specific part of the compiler
diff --git a/src/pystencils/nbackend/kernelcreation/typification.py b/src/pystencils/nbackend/kernelcreation/typification.py
index 4a228e9e24d64a32853d03aea18aed3050f7ea76..65263cbe8a8b8e9fecb8055976ab23295037ffca 100644
--- a/src/pystencils/nbackend/kernelcreation/typification.py
+++ b/src/pystencils/nbackend/kernelcreation/typification.py
@@ -1,17 +1,50 @@
+from __future__ import annotations
+
+from typing import TypeVar
+
 import pymbolic.primitives as pb
 from pymbolic.mapper import Mapper
 
 from .context import KernelCreationContext
 from ..types import PsAbstractType
 from ..typed_expressions import PsTypedVariable
+from ..ast import PsAstNode, PsExpression, PsAssignment
+
+
+class TypificationException(Exception):
+    """Indicates a fatal error during typification."""
+
+
+NodeT = TypeVar("NodeT", bound=PsAstNode)
 
 
 class Typifier(Mapper):
     def __init__(self, ctx: KernelCreationContext):
         self._ctx = ctx
 
-    def __call__(self, expr: pb.Expression) -> tuple[pb.Expression, PsAbstractType]:
-        return self.rec(expr)
+    def __call__(self, node: NodeT) -> NodeT:
+        match node:
+            case PsExpression(expr):
+                node.expression, _ = self.rec(expr)
+
+            case PsAssignment(lhs, rhs):
+                lhs, lhs_dtype = self.rec(lhs)
+                rhs, rhs_dtype = self.rec(rhs)
+                if lhs_dtype != rhs_dtype:
+                    #   todo: (optional) automatic cast insertion?
+                    raise TypificationException(
+                        "Mismatched types in assignment: \n"
+                        f"    {lhs} <- {rhs}\n"
+                        f"    dtype(lhs) = {lhs_dtype}\n"
+                        f"    dtype(rhs) = {rhs_dtype}\n"
+                    )
+                node.lhs = lhs
+                node.rhs = rhs
+
+            case unknown:
+                raise NotImplementedError(f"Don't know how to typify {unknown}")
+            
+        return node
 
     def map_variable(self, var: pb.Variable) -> tuple[pb.Expression, PsAbstractType]:
         dtype = NotImplemented  # determine variable type