diff --git a/gem/gem.py b/gem/gem.py
index c37e4352..ded6bf65 100644
--- a/gem/gem.py
+++ b/gem/gem.py
@@ -56,6 +56,9 @@ def __call__(self, *args, **kwargs):
         if not hasattr(obj, 'free_indices'):
             obj.free_indices = unique(chain(*[c.free_indices
                                               for c in obj.children]))
+        # Set dtype if not set already.
+        if not hasattr(obj, 'dtype'):
+            obj.dtype = obj.inherit_dtype_from_children(obj.children)
 
         return obj
 
@@ -63,7 +66,7 @@ def __call__(self, *args, **kwargs):
 class Node(NodeBase, metaclass=NodeMeta):
     """Abstract GEM node class."""
 
-    __slots__ = ('free_indices',)
+    __slots__ = ('free_indices', 'dtype')
 
     def is_equal(self, other):
         """Common subexpression eliminating equality predicate.
@@ -153,16 +156,46 @@ def __mod__(self, other):
     def __rmod__(self, other):
         return as_gem_uint(other).__mod__(self)
 
+    @staticmethod
+    def inherit_dtype_from_children(children):
+        if any(c.dtype is None for c in children):
+            # Set dtype = None will let _assign_dtype()
+            # assign the default dtype for this node later.
+            return
+        else:
+            return numpy.result_type(*(c.dtype for c in children))
+
 
 class Terminal(Node):
     """Abstract class for terminal GEM nodes."""
 
-    __slots__ = ()
+    __slots__ = ('_dtype',)
 
     children = ()
 
     is_equal = NodeBase.is_equal
 
+    @property
+    def dtype(self):
+        """dtype of the node.
+
+        We only need to set dtype (or _dtype) on terminal nodes, and
+        other nodes inherit dtype from their children.
+
+        Currently dtype is significant only for nodes under index DAGs
+        (DAGs underneath `VariableIndex`s representing indices), and
+        `VariableIndex` checks if the dtype of the node that it wraps is
+        of uint_type. _assign_dtype() will then assign uint_type to those nodes.
+
+        dtype can be `None` otherwise, and _assign_dtype() will assign
+        the default dtype to those nodes.
+
+        """
+        if hasattr(self, '_dtype'):
+            return self._dtype
+        else:
+            raise AttributeError(f"Must set _dtype on terminal node, {type(self)}")
+
 
 class Scalar(Node):
     """Abstract class for scalar-valued GEM nodes."""
@@ -181,6 +214,7 @@ class Failure(Terminal):
     def __init__(self, shape, exception):
         self.shape = shape
         self.exception = exception
+        self._dtype = None
 
 
 class Constant(Terminal):
@@ -190,8 +224,7 @@ class Constant(Terminal):
      - array: numpy array of values
      - value: float or complex value (scalars only)
     """
-    __slots__ = ('dtype',)
-    __back__ = ('dtype',)
+    pass
 
 
 class Zero(Constant):
@@ -199,15 +232,16 @@ class Zero(Constant):
 
     __slots__ = ('shape',)
     __front__ = ('shape',)
+    __back__ = ('dtype',)
 
-    def __init__(self, shape=(), dtype=float):
+    def __init__(self, shape=(), dtype=None):
         self.shape = shape
-        self.dtype = dtype
+        self._dtype = dtype
 
     @property
     def value(self):
         assert not self.shape
-        return numpy.array(0, dtype=self.dtype).item()
+        return numpy.array(0, dtype=self.dtype or float).item()
 
 
 class Identity(Constant):
@@ -215,10 +249,11 @@ class Identity(Constant):
 
     __slots__ = ('dim',)
     __front__ = ('dim',)
+    __back__ = ('dtype',)
 
-    def __init__(self, dim, dtype=float):
+    def __init__(self, dim, dtype=None):
         self.dim = dim
-        self.dtype = dtype
+        self._dtype = dtype
 
     @property
     def shape(self):
@@ -234,6 +269,7 @@ class Literal(Constant):
 
     __slots__ = ('array',)
     __front__ = ('array',)
+    __back__ = ('dtype',)
 
     def __new__(cls, array, dtype=None):
         array = asarray(array)
@@ -245,14 +281,12 @@ def __init__(self, array, dtype=None):
             # Assume float or complex.
             try:
                 self.array = array.astype(float, casting="safe")
-                self.dtype = float
             except TypeError:
                 self.array = array.astype(complex)
-                self.dtype = complex
         else:
             # Can be int, etc.
             self.array = array.astype(dtype)
-            self.dtype = dtype
+        self._dtype = self.array.dtype
 
     def is_equal(self, other):
         if type(self) is not type(other):
@@ -277,13 +311,14 @@ def shape(self):
 class Variable(Terminal):
     """Symbolic variable tensor"""
 
-    __slots__ = ('name', 'shape', 'dtype')
-    __front__ = ('name', 'shape', 'dtype')
+    __slots__ = ('name', 'shape')
+    __front__ = ('name', 'shape')
+    __back__ = ('dtype',)
 
     def __init__(self, name, shape, dtype=None):
         self.name = name
         self.shape = shape
-        self.dtype = dtype
+        self._dtype = dtype
 
 
 class Sum(Scalar):
@@ -300,8 +335,7 @@ def __new__(cls, a, b):
             return a
 
         if isinstance(a, Constant) and isinstance(b, Constant):
-            dtype = numpy.result_type(a.dtype, b.dtype)
-            return Literal(a.value + b.value, dtype=dtype)
+            return Literal(a.value + b.value, dtype=Node.inherit_dtype_from_children([a, b]))
 
         self = super(Sum, cls).__new__(cls)
         self.children = a, b
@@ -325,8 +359,7 @@ def __new__(cls, a, b):
             return a
 
         if isinstance(a, Constant) and isinstance(b, Constant):
-            dtype = numpy.result_type(a.dtype, b.dtype)
-            return Literal(a.value * b.value, dtype=dtype)
+            return Literal(a.value * b.value, dtype=Node.inherit_dtype_from_children([a, b]))
 
         self = super(Product, cls).__new__(cls)
         self.children = a, b
@@ -350,8 +383,7 @@ def __new__(cls, a, b):
             return a
 
         if isinstance(a, Constant) and isinstance(b, Constant):
-            dtype = numpy.result_type(a.dtype, b.dtype)
-            return Literal(a.value / b.value, dtype=dtype)
+            return Literal(a.value / b.value, dtype=Node.inherit_dtype_from_children([a, b]))
 
         self = super(Division, cls).__new__(cls)
         self.children = a, b
@@ -364,18 +396,17 @@ class FloorDiv(Scalar):
     def __new__(cls, a, b):
         assert not a.shape
         assert not b.shape
-        # TODO: Attach dtype property to Node and check that
-        #       numpy.result_dtype(a.dtype, b.dtype) is uint type.
-        #       dtype is currently attached only to {Constant, Variable}.
+        dtype = Node.inherit_dtype_from_children([a, b])
+        if dtype is not uint_type:
+            raise ValueError(f"dtype ({dtype}) is not unit_type ({uint_type})")
         # Constant folding
         if isinstance(b, Zero):
             raise ValueError("division by zero")
         if isinstance(a, Zero):
-            return Zero(dtype=a.dtype)
+            return Zero(dtype=dtype)
         if isinstance(b, Constant) and b.value == 1:
             return a
         if isinstance(a, Constant) and isinstance(b, Constant):
-            dtype = numpy.result_type(a.dtype, b.dtype)
             return Literal(a.value // b.value, dtype=dtype)
         self = super(FloorDiv, cls).__new__(cls)
         self.children = a, b
@@ -388,18 +419,17 @@ class Remainder(Scalar):
     def __new__(cls, a, b):
         assert not a.shape
         assert not b.shape
-        # TODO: Attach dtype property to Node and check that
-        #       numpy.result_dtype(a.dtype, b.dtype) is uint type.
-        #       dtype is currently attached only to {Constant, Variable}.
+        dtype = Node.inherit_dtype_from_children([a, b])
+        if dtype is not uint_type:
+            raise ValueError(f"dtype ({dtype}) is not uint_type ({uint_type})")
         # Constant folding
         if isinstance(b, Zero):
             raise ValueError("division by zero")
         if isinstance(a, Zero):
-            return Zero(dtype=a.dtype)
+            return Zero(dtype=dtype)
         if isinstance(b, Constant) and b.value == 1:
-            return Zero(dtype=b.dtype)
+            return Zero(dtype=dtype)
         if isinstance(a, Constant) and isinstance(b, Constant):
-            dtype = numpy.result_type(a.dtype, b.dtype)
             return Literal(a.value % b.value, dtype=dtype)
         self = super(Remainder, cls).__new__(cls)
         self.children = a, b
@@ -412,18 +442,16 @@ class Power(Scalar):
     def __new__(cls, base, exponent):
         assert not base.shape
         assert not exponent.shape
+        dtype = Node.inherit_dtype_from_children([base, exponent])
 
         # Constant folding
         if isinstance(base, Zero):
-            dtype = numpy.result_type(base.dtype, exponent.dtype)
             if isinstance(exponent, Zero):
                 raise ValueError("cannot solve 0^0")
             return Zero(dtype=dtype)
         elif isinstance(exponent, Zero):
-            dtype = numpy.result_type(base.dtype, exponent.dtype)
             return Literal(1, dtype=dtype)
         elif isinstance(base, Constant) and isinstance(exponent, Constant):
-            dtype = numpy.result_type(base.dtype, exponent.dtype)
             return Literal(base.value ** exponent.value, dtype=dtype)
 
         self = super(Power, cls).__new__(cls)
@@ -483,6 +511,7 @@ def __init__(self, op, a, b):
 
         self.operator = op
         self.children = a, b
+        self.dtype = None  # Do not inherit dtype from children.
 
 
 class LogicalNot(Scalar):
@@ -529,6 +558,7 @@ def __new__(cls, condition, then, else_):
         self = super(Conditional, cls).__new__(cls)
         self.children = condition, then, else_
         self.shape = then.shape
+        self.dtype = Node.inherit_dtype_from_children([then, else_])
         return self
 
 
@@ -591,6 +621,8 @@ class VariableIndex(IndexBase):
     def __init__(self, expression):
         assert isinstance(expression, Node)
         assert not expression.shape
+        if expression.dtype is not uint_type:
+            raise ValueError(f"expression.dtype ({expression.dtype}) is not uint_type ({uint_type})")
         self.expression = expression
 
     def __eq__(self, other):
@@ -846,6 +878,7 @@ class ListTensor(Node):
     def __new__(cls, array):
         array = asarray(array)
         assert numpy.prod(array.shape)
+        dtype = Node.inherit_dtype_from_children(tuple(array.flat))
 
         # Handle children with shape
         child_shape = array.flat[0].shape
@@ -861,7 +894,7 @@ def __new__(cls, array):
 
         # Constant folding
         if all(isinstance(elem, Constant) for elem in array.flat):
-            return Literal(numpy.vectorize(attrgetter('value'))(array))
+            return Literal(numpy.vectorize(attrgetter('value'))(array), dtype=dtype)
 
         self = super(ListTensor, cls).__new__(cls)
         self.array = array
@@ -907,9 +940,9 @@ class Concatenate(Node):
     __slots__ = ('children',)
 
     def __new__(cls, *children):
+        dtype = Node.inherit_dtype_from_children(children)
         if all(isinstance(child, Zero) for child in children):
             size = int(sum(numpy.prod(child.shape, dtype=int) for child in children))
-            dtype = numpy.result_type(*(child.dtype for child in children))
             return Zero((size,), dtype=dtype)
 
         self = super(Concatenate, cls).__new__(cls)
@@ -924,8 +957,9 @@ def shape(self):
 class Delta(Scalar, Terminal):
     __slots__ = ('i', 'j')
     __front__ = ('i', 'j')
+    __back__ = ('dtype',)
 
-    def __new__(cls, i, j):
+    def __new__(cls, i, j, dtype=None):
         assert isinstance(i, IndexBase)
         assert isinstance(j, IndexBase)
 
@@ -948,6 +982,7 @@ def __new__(cls, i, j):
             elif isinstance(index, VariableIndex):
                 raise NotImplementedError("Can not make Delta with VariableIndex")
         self.free_indices = tuple(unique(free_indices))
+        self._dtype = dtype
         return self
 
 
diff --git a/tests/test_pickle_gem.py b/tests/test_pickle_gem.py
index 73e39cac..beb101f9 100644
--- a/tests/test_pickle_gem.py
+++ b/tests/test_pickle_gem.py
@@ -6,7 +6,7 @@
 
 @pytest.mark.parametrize('protocol', range(3))
 def test_pickle_gem(protocol):
-    f = gem.VariableIndex(gem.Indexed(gem.Variable('facet', (2,)), (1,)))
+    f = gem.VariableIndex(gem.Indexed(gem.Variable('facet', (2,), dtype=gem.uint_type), (1,)))
     q = gem.Index()
     r = gem.Index()
     _1 = gem.Indexed(gem.Literal(numpy.random.rand(3, 6, 8)), (f, q, r))
diff --git a/tsfc/kernel_interface/firedrake_loopy.py b/tsfc/kernel_interface/firedrake_loopy.py
index cc35a7c5..0969854c 100644
--- a/tsfc/kernel_interface/firedrake_loopy.py
+++ b/tsfc/kernel_interface/firedrake_loopy.py
@@ -85,11 +85,11 @@ def __init__(self, scalar_type, interior_facet=False):
 
         # Cell orientation
         if self.interior_facet:
-            cell_orientations = gem.Variable("cell_orientations", (2,))
+            cell_orientations = gem.Variable("cell_orientations", (2,), dtype=gem.uint_type)
             self._cell_orientations = (gem.Indexed(cell_orientations, (0,)),
                                        gem.Indexed(cell_orientations, (1,)))
         else:
-            cell_orientations = gem.Variable("cell_orientations", (1,))
+            cell_orientations = gem.Variable("cell_orientations", (1,), dtype=gem.uint_type)
             self._cell_orientations = (gem.Indexed(cell_orientations, (0,)),)
 
     def _coefficient(self, coefficient, name):
@@ -257,12 +257,12 @@ def __init__(self, integral_data_info, scalar_type,
 
         # Facet number
         if integral_type in ['exterior_facet', 'exterior_facet_vert']:
-            facet = gem.Variable('facet', (1,))
+            facet = gem.Variable('facet', (1,), dtype=gem.uint_type)
             self._entity_number = {None: gem.VariableIndex(gem.Indexed(facet, (0,)))}
             facet_orientation = gem.Variable('facet_orientation', (1,), dtype=gem.uint_type)
             self._entity_orientation = {None: gem.OrientationVariableIndex(gem.Indexed(facet_orientation, (0,)))}
         elif integral_type in ['interior_facet', 'interior_facet_vert']:
-            facet = gem.Variable('facet', (2,))
+            facet = gem.Variable('facet', (2,), dtype=gem.uint_type)
             self._entity_number = {
                 '+': gem.VariableIndex(gem.Indexed(facet, (0,))),
                 '-': gem.VariableIndex(gem.Indexed(facet, (1,)))
diff --git a/tsfc/loopy.py b/tsfc/loopy.py
index d19da423..08efaede 100644
--- a/tsfc/loopy.py
+++ b/tsfc/loopy.py
@@ -47,7 +47,7 @@ def _assign_dtype(expression, self):
 
 @_assign_dtype.register(gem.Terminal)
 def _assign_dtype_terminal(expression, self):
-    return {self.scalar_type}
+    return {expression.dtype or self.scalar_type}
 
 
 @_assign_dtype.register(gem.Variable)
@@ -59,7 +59,7 @@ def _assign_dtype_variable(expression, self):
 @_assign_dtype.register(gem.Identity)
 @_assign_dtype.register(gem.Delta)
 def _assign_dtype_real(expression, self):
-    return {self.real_type}
+    return {expression.dtype or self.real_type}
 
 
 @_assign_dtype.register(gem.Literal)
@@ -70,15 +70,15 @@ def _assign_dtype_identity(expression, self):
 @_assign_dtype.register(gem.Power)
 def _assign_dtype_power(expression, self):
     # Conservative
-    return {self.scalar_type}
+    return {expression.dtype or self.scalar_type}
 
 
 @_assign_dtype.register(gem.MathFunction)
 def _assign_dtype_mathfunction(expression, self):
     if expression.name in {"abs", "real", "imag"}:
-        return {self.real_type}
+        return {expression.dtype or self.real_type}
     elif expression.name == "sqrt":
-        return {self.scalar_type}
+        return {expression.dtype or self.scalar_type}
     else:
         return set.union(*map(self, expression.children))
 
@@ -87,7 +87,7 @@ def _assign_dtype_mathfunction(expression, self):
 @_assign_dtype.register(gem.MaxValue)
 def _assign_dtype_minmax(expression, self):
     # UFL did correctness checking
-    return {self.real_type}
+    return {expression.dtype or self.real_type}
 
 
 @_assign_dtype.register(gem.Conditional)
@@ -100,7 +100,7 @@ def _assign_dtype_conditional(expression, self):
 @_assign_dtype.register(gem.LogicalAnd)
 @_assign_dtype.register(gem.LogicalOr)
 def _assign_dtype_logical(expression, self):
-    return {numpy.int8}
+    return {expression.dtype or numpy.int8}
 
 
 def assign_dtypes(expressions, scalar_type):