diff --git a/cirq/__init__.py b/cirq/__init__.py
index 38d20130f5f..342a95f706d 100644
--- a/cirq/__init__.py
+++ b/cirq/__init__.py
@@ -30,6 +30,7 @@
 
 from cirq.circuits import (
     Circuit,
+    CircuitDag,
     DropEmptyMoments,
     DropNegligible,
     ExpandComposite,
diff --git a/cirq/circuits/__init__.py b/cirq/circuits/__init__.py
index ce212a0e6fb..2ccbca0a9df 100644
--- a/cirq/circuits/__init__.py
+++ b/cirq/circuits/__init__.py
@@ -23,6 +23,9 @@
 from cirq.circuits.circuit import (
     Circuit,
 )
+from cirq.circuits.circuit_dag import (
+    CircuitDag,
+)
 from cirq.circuits.drop_empty_moments import (
     DropEmptyMoments,
 )
diff --git a/cirq/circuits/circuit_dag.py b/cirq/circuits/circuit_dag.py
new file mode 100644
index 00000000000..590d04afc4a
--- /dev/null
+++ b/cirq/circuits/circuit_dag.py
@@ -0,0 +1,154 @@
+# Copyright 2018 The ops Developers
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     https://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+from typing import Any, Callable, Generic, Iterator, TypeVar
+
+import networkx
+
+from cirq import ops, devices
+from cirq.circuits import circuit
+
+
+T = TypeVar('T')
+
+class Unique(Generic[T]):
+    """A wrapper for a value that doesn't compare equal to other instances.
+
+    For example: 5 == 5 but Unique(5) != Unique(5).
+
+    Unique is used by CircuitDag to wrap operations because nodes in a graph
+    are considered the same node if they compare equal to each other.  X(q0)
+    in one moment of a Circuit and X(q0) in another moment of the Circuit are
+    wrapped by Unique(X(q0)) so they are distinct nodes in the graph.
+    """
+    def __init__(self, val: T) -> None:
+        self.val = val
+
+    def __repr__(self):
+        return 'Unique({}, {!r})'.format(id(self), self.val)
+
+
+def _disjoint_qubits(op1: ops.Operation, op2: ops.Operation) -> bool:
+    """Returns true only if the operations have qubits in common."""
+    return not set(op1.qubits) & set(op2.qubits)
+
+
+class CircuitDag(networkx.DiGraph):
+    """A representation of a Circuit as a directed acyclic graph.
+
+    Nodes of the graph are instances of Unique containing each operation of a
+    circuit.
+
+    Edges of the graph are tuples of nodes.  Each edge specifies a required
+    application order between two operations.  The first must be applied before
+    the second.
+
+    The graph is maximalist (transitive completion).
+    """
+
+    disjoint_qubits = staticmethod(_disjoint_qubits)
+
+    def __init__(self,
+                 can_reorder: Callable[[ops.Operation, ops.Operation],
+                                       bool] = _disjoint_qubits,
+                 incoming_graph_data: Any = None,
+                 device: devices.Device = devices.UnconstrainedDevice
+                 ) -> None:
+        """Initializes a CircuitDag.
+
+        Args:
+            can_reorder: A predicate that determines if two operations may be
+                reordered.  Graph edges are created for pairs of operations
+                where this returns False.
+
+                The default predicate allows reordering only when the operations
+                don't share common qubits.
+            incoming_graph_data: Data in initialize the graph.  This can be any
+                value supported by networkx.DiGraph() e.g. an edge list or
+                another graph.
+            device: Hardware that the circuit should be able to run on.
+        """
+        super().__init__(incoming_graph_data)
+        self.can_reorder = can_reorder
+        self.device = device
+
+    @staticmethod
+    def make_node(op: ops.Operation) -> Unique:
+        return Unique(op)
+
+    @staticmethod
+    def from_circuit(circuit: circuit.Circuit,
+                     can_reorder: Callable[[ops.Operation, ops.Operation],
+                                           bool] = _disjoint_qubits
+                     ) -> 'CircuitDag':
+        return CircuitDag.from_ops(circuit.all_operations(),
+                                   can_reorder=can_reorder,
+                                   device=circuit.device)
+
+    @staticmethod
+    def from_ops(*operations: ops.OP_TREE,
+                 can_reorder: Callable[[ops.Operation, ops.Operation],
+                                       bool] = _disjoint_qubits,
+                 device: devices.Device = devices.UnconstrainedDevice
+                 ) -> 'CircuitDag':
+        dag = CircuitDag(can_reorder=can_reorder, device=device)
+        for op in ops.flatten_op_tree(operations):
+            dag.append(op)
+        return dag
+
+    def append(self, op: ops.Operation) -> None:
+        new_node = self.make_node(op)
+        self.add_edges_from([(node, new_node)
+                             for node in self.nodes
+                             if not self.can_reorder(node.val, new_node.val)])
+        self.add_node(new_node)
+
+    def all_operations(self) -> Iterator[ops.Operation]:
+        if not self.nodes:
+            return
+        g = self.copy()
+
+        def get_root_node(some_node: Unique[ops.Operation]
+                          ) -> Unique[ops.Operation]:
+            pred = g.pred
+            while pred[some_node]:
+                some_node = next(iter(pred[some_node]))
+            return some_node
+
+        def get_first_node() -> Unique[ops.Operation]:
+            return get_root_node(next(iter(g.nodes)))
+
+        def get_next_node(succ: networkx.classes.coreviews.AtlasView
+                          ) -> Unique[ops.Operation]:
+            if succ:
+                return get_root_node(next(iter(succ)))
+            else:
+                return get_first_node()
+
+        node = get_first_node()
+        while True:
+            yield node.val
+            succ = g.succ[node]
+            g.remove_node(node)
+
+            if not g.nodes:
+                return
+
+            node = get_next_node(succ)
+
+    def to_circuit(self) -> circuit.Circuit:
+        return circuit.Circuit.from_ops(
+                    self.all_operations(),
+                    strategy=circuit.InsertStrategy.EARLIEST,
+                    device=self.device)
diff --git a/cirq/circuits/circuit_dag_test.py b/cirq/circuits/circuit_dag_test.py
new file mode 100644
index 00000000000..f4453d56b30
--- /dev/null
+++ b/cirq/circuits/circuit_dag_test.py
@@ -0,0 +1,170 @@
+# Copyright 2018 The Cirq Developers
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     https://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+import networkx
+
+import cirq
+
+
+def test_wrapper_eq():
+    q0, q1 = cirq.LineQubit.range(2)
+    eq = cirq.testing.EqualsTester()
+    eq.add_equality_group(cirq.CircuitDag.make_node(cirq.X(q0)))
+    eq.add_equality_group(cirq.CircuitDag.make_node(cirq.X(q0)))
+    eq.add_equality_group(cirq.CircuitDag.make_node(cirq.Y(q0)))
+    eq.add_equality_group(cirq.CircuitDag.make_node(cirq.X(q1)))
+
+
+def test_wrapper_repr():
+    q0 = cirq.LineQubit(0)
+
+    node = cirq.CircuitDag.make_node(cirq.X(q0))
+    assert (repr(node) ==
+            'Unique(' + str(id(node)) + ', GateOperation(X, (LineQubit(0),)))')
+
+
+def test_init():
+    dag = cirq.CircuitDag()
+    assert networkx.dag.is_directed_acyclic_graph(dag)
+    assert list(dag.nodes) == []
+    assert list(dag.edges) == []
+
+
+def test_append():
+    q0 = cirq.LineQubit(0)
+    dag = cirq.CircuitDag()
+    dag.append(cirq.X(q0))
+    dag.append(cirq.Y(q0))
+    print(dag.edges)
+    assert networkx.dag.is_directed_acyclic_graph(dag)
+    assert len(dag.nodes) == 2
+    assert ([(n1.val, n2.val) for n1, n2 in dag.edges] ==
+            [(cirq.X(q0), cirq.Y(q0))])
+
+
+def test_two_identical_ops():
+    q0 = cirq.LineQubit(0)
+    dag = cirq.CircuitDag()
+    dag.append(cirq.X(q0))
+    dag.append(cirq.Y(q0))
+    dag.append(cirq.X(q0))
+    assert networkx.dag.is_directed_acyclic_graph(dag)
+    assert len(dag.nodes) == 3
+    assert (set((n1.val, n2.val) for n1, n2 in dag.edges) ==
+            set(((cirq.X(q0), cirq.Y(q0)),
+                 (cirq.X(q0), cirq.X(q0)),
+                 (cirq.Y(q0), cirq.X(q0)))))
+
+
+def test_from_ops():
+    q0 = cirq.LineQubit(0)
+    dag = cirq.CircuitDag.from_ops(
+        cirq.X(q0),
+        cirq.Y(q0))
+    assert networkx.dag.is_directed_acyclic_graph(dag)
+    assert len(dag.nodes) == 2
+    assert ([(n1.val, n2.val) for n1, n2 in dag.edges] ==
+            [(cirq.X(q0), cirq.Y(q0))])
+
+
+def test_from_circuit():
+    q0 = cirq.LineQubit(0)
+    circuit = cirq.Circuit.from_ops(
+        cirq.X(q0),
+        cirq.Y(q0))
+    dag = cirq.CircuitDag.from_circuit(circuit)
+    assert networkx.dag.is_directed_acyclic_graph(dag)
+    assert len(dag.nodes) == 2
+    assert ([(n1.val, n2.val) for n1, n2 in dag.edges] ==
+            [(cirq.X(q0), cirq.Y(q0))])
+
+
+def test_from_circuit_with_device():
+    q0 = cirq.GridQubit(5, 5)
+    circuit = cirq.Circuit.from_ops(
+        cirq.X(q0),
+        cirq.Y(q0),
+        device=cirq.google.Bristlecone)
+    dag = cirq.CircuitDag.from_circuit(circuit)
+    assert networkx.dag.is_directed_acyclic_graph(dag)
+    assert dag.device == circuit.device
+    assert len(dag.nodes) == 2
+    assert ([(n1.val, n2.val) for n1, n2 in dag.edges] ==
+            [(cirq.X(q0), cirq.Y(q0))])
+
+
+def test_to_empty_circuit():
+    circuit = cirq.Circuit()
+    dag = cirq.CircuitDag.from_circuit(circuit)
+    assert networkx.dag.is_directed_acyclic_graph(dag)
+    assert circuit == dag.to_circuit()
+
+
+def test_to_circuit():
+    q0 = cirq.LineQubit(0)
+    circuit = cirq.Circuit.from_ops(
+        cirq.X(q0),
+        cirq.Y(q0))
+    dag = cirq.CircuitDag.from_circuit(circuit)
+
+    assert networkx.dag.is_directed_acyclic_graph(dag)
+    # Only one possible output circuit for this simple case
+    assert circuit == dag.to_circuit()
+
+    cirq.testing.assert_allclose_up_to_global_phase(
+        circuit.to_unitary_matrix(),
+        dag.to_circuit().to_unitary_matrix(),
+        atol=1e-7)
+
+
+def test_larger_circuit():
+    q0, q1, q2, q3 = cirq.google.Bristlecone.col(5)[:4]
+    # This circuit does not have CZ gates on adjacent qubits because the order
+    # dag.to_circuit() would append them is non-deterministic.
+    circuit = cirq.Circuit.from_ops(
+        cirq.X(q0),
+        cirq.CZ(q1, q2),
+        cirq.CZ(q0, q1),
+        cirq.Y(q0),
+        cirq.Z(q0),
+        cirq.CZ(q1, q2),
+        cirq.X(q0),
+        cirq.Y(q0),
+        cirq.CZ(q0, q1),
+        cirq.T(q3),
+        strategy=cirq.InsertStrategy.EARLIEST,
+        device=cirq.google.Bristlecone)
+
+    dag = cirq.CircuitDag.from_circuit(circuit)
+
+    assert networkx.dag.is_directed_acyclic_graph(dag)
+    assert circuit.device == dag.to_circuit().device
+    # Operation order within a moment is non-deterministic
+    # but text diagrams still look the same.
+    assert (circuit.to_text_diagram() ==
+            dag.to_circuit().to_text_diagram() ==
+"""
+(0, 5): ───X───@───Y───Z───X───Y───@───
+               │                   │
+(1, 5): ───@───@───@───────────────@───
+           │       │
+(2, 5): ───@───────@───────────────────
+
+(3, 5): ───T───────────────────────────
+""".strip())
+
+    cirq.testing.assert_allclose_up_to_global_phase(
+        circuit.to_unitary_matrix(),
+        dag.to_circuit().to_unitary_matrix(),
+        atol=1e-7)
diff --git a/continuous-integration/mypy.ini b/continuous-integration/mypy.ini
index 901e05669ea..ad908593530 100644
--- a/continuous-integration/mypy.ini
+++ b/continuous-integration/mypy.ini
@@ -5,6 +5,6 @@ follow_imports = silent
 ignore_missing_imports = true
 
 # 3rd-party libs for which we don't have stubs
-[mypy-absl.*,apiclient.*,google.protobuf.*,matplotlib.*,multiprocessing.dummy,numpy.*,oauth2client.*,pytest.*,scipy.*,sortedcontainers.*,setuptools.*,pylatex.*]
+[mypy-absl.*,apiclient.*,google.protobuf.*,matplotlib.*,multiprocessing.dummy,numpy.*,oauth2client.*,pytest.*,scipy.*,sortedcontainers.*,setuptools.*,pylatex.*,networkx.*]
 follow_imports = silent
 ignore_missing_imports = true
diff --git a/python2.7-runtime-requirements.txt b/python2.7-runtime-requirements.txt
index 16705664124..70614690a2a 100644
--- a/python2.7-runtime-requirements.txt
+++ b/python2.7-runtime-requirements.txt
@@ -1,5 +1,6 @@
 google-api-python-client~=1.6
 matplotlib~=2.1
+networkx~=2.1
 numpy~=1.13
 protobuf~=3.5
 sortedcontainers~=1.5
diff --git a/runtime-requirements.txt b/runtime-requirements.txt
index 7ca5348136c..28aaebabba5 100644
--- a/runtime-requirements.txt
+++ b/runtime-requirements.txt
@@ -1,5 +1,6 @@
 google-api-python-client~=1.6
 matplotlib~=2.2
+networkx~=2.1
 numpy~=1.14
 protobuf~=3.5
 requests~=2.18