diff --git a/tests/interpreters/test_memref_interpreter.py b/tests/interpreters/test_memref_interpreter.py
new file mode 100644
index 0000000000..41ac972797
--- /dev/null
+++ b/tests/interpreters/test_memref_interpreter.py
@@ -0,0 +1,50 @@
+import pytest
+
+from xdsl.dialects import arith, memref
+from xdsl.dialects.builtin import IndexType, ModuleOp, i32
+from xdsl.interpreter import Interpreter
+from xdsl.interpreters.arith import ArithFunctions
+from xdsl.interpreters.memref import MemrefFunctions, MemrefValue
+from xdsl.interpreters.shaped_array import ShapedArray
+from xdsl.utils.exceptions import InterpretationError
+
+interpreter = Interpreter(ModuleOp([]))
+interpreter.register_implementations(ArithFunctions())
+interpreter.register_implementations(MemrefFunctions())
+
+index = IndexType()
+
+
+def test_functions():
+    alloc_op = memref.Alloc.get(i32, None, (2, 3))
+    zero_op = arith.Constant.from_int_and_width(0, index)
+    one_op = arith.Constant.from_int_and_width(1, index)
+    two_op = arith.Constant.from_int_and_width(2, index)
+    forty_two_op = arith.Constant.from_int_and_width(42, 32)
+    store_op = memref.Store.get(forty_two_op, alloc_op, (zero_op, one_op))
+    load_42_op = memref.Load.get(alloc_op, (zero_op, one_op))
+    load_undef_op = memref.Load.get(alloc_op, (zero_op, two_op))
+    dealloc_op = memref.Dealloc.get(alloc_op)
+
+    (shaped_array,) = interpreter.run_op(alloc_op, ())
+    v = MemrefValue.Allocated
+    assert shaped_array == ShapedArray([v, v, v, v, v, v], [2, 3])
+    (zero,) = interpreter.run_op(zero_op, ())
+    (one,) = interpreter.run_op(one_op, ())
+    (two,) = interpreter.run_op(two_op, ())
+    (forty_two_0,) = interpreter.run_op(forty_two_op, ())
+    store_res = interpreter.run_op(store_op, (forty_two_0, shaped_array, zero, one))
+    assert store_res == ()
+    (forty_two_1,) = interpreter.run_op(load_42_op, (shaped_array, zero, one))
+    assert forty_two_1 == 42
+
+    with pytest.raises(InterpretationError) as e:
+        interpreter.run_op(load_undef_op, (shaped_array, zero, two))
+        e.match("uninitialized")
+
+    dealloc_res = interpreter.run_op(dealloc_op, (shaped_array,))
+    assert dealloc_res == ()
+
+    with pytest.raises(InterpretationError) as e:
+        interpreter.run_op(load_undef_op, (shaped_array, zero, one))
+        e.match("deallocated")
diff --git a/xdsl/interpreters/memref.py b/xdsl/interpreters/memref.py
new file mode 100644
index 0000000000..ffcb5fc105
--- /dev/null
+++ b/xdsl/interpreters/memref.py
@@ -0,0 +1,69 @@
+from enum import Enum
+from math import prod
+from typing import Any, cast
+
+from xdsl.dialects import memref
+from xdsl.interpreter import Interpreter, InterpreterFunctions, impl, register_impls
+from xdsl.interpreters.shaped_array import ShapedArray
+from xdsl.ir.core import Attribute
+from xdsl.utils.exceptions import InterpretationError
+
+
+class MemrefValue(Enum):
+    Allocated = 1
+    Deallocated = 2
+
+
+@register_impls
+class MemrefFunctions(InterpreterFunctions):
+    @impl(memref.Alloc)
+    def run_alloc(self, interpreter: Interpreter, op: memref.Alloc, args: tuple[()]):
+        memref_typ = cast(memref.MemRefType[Attribute], op.memref.type)
+
+        shape = memref_typ.get_shape()
+        size = prod(shape)
+        data = [MemrefValue.Allocated] * size
+
+        shaped_array = ShapedArray(data, list(shape))
+        return (shaped_array,)
+
+    @impl(memref.Dealloc)
+    def run_dealloc(
+        self, interpreter: Interpreter, op: memref.Dealloc, args: tuple[Any, ...]
+    ):
+        (shaped_array,) = args
+        for i in range(len(shaped_array.data)):
+            shaped_array.data[i] = MemrefValue.Deallocated
+        return ()
+
+    @impl(memref.Store)
+    def run_store(
+        self, interpreter: Interpreter, op: memref.Store, args: tuple[Any, ...]
+    ):
+        value, memref, *indices = args
+
+        memref = cast(ShapedArray[Any], memref)
+
+        indices = tuple(indices)
+        memref.store(indices, value)
+
+        return ()
+
+    @impl(memref.Load)
+    def run_load(
+        self, interpreter: Interpreter, op: memref.Load, args: tuple[Any, ...]
+    ):
+        shaped_array, *indices = args
+
+        shaped_array = cast(ShapedArray[Any], shaped_array)
+
+        indices = tuple(indices)
+        value = shaped_array.load(indices)
+
+        if isinstance(value, MemrefValue):
+            state = "uninitialized" if value == MemrefValue.Allocated else "deallocated"
+            raise InterpretationError(
+                f"Cannot load {state} value from memref {shaped_array}"
+            )
+
+        return (value,)