The effort to discover a quantum theory of gravity is motivated by the need to reconcile the incompatibility between quantum theory and general relativity. Here, we present an alternative approach by constructing a consistent theory of classical gravity coupled to quantum field theory. The dynamics is linear in the density matrix, completely positive and trace preserving, and reduces to Einstein's theory of general relativity in the classical limit. Consequently, the dynamics doesn't suffer from the pathologies of the semiclassical theory based on expectation values. The assumption that general relativity is classical necessarily modifies the dynamical laws of quantum mechanics -- the theory must be fundamentally stochastic in both the metric degrees of freedom and in the quantum matter fields. This allows it to evade several no-go theorems purporting to forbid classical-quantum interactions. The measurement postulate of quantum mechanics is not needed -- the interaction of the quantum degrees of freedom with classical space-time necessarily causes decoherence in the quantum system. We first derive the general form of classical-quantum dynamics and consider realisations which have as its limit deterministic classical Hamiltonian evolution. The formalism is then applied to quantum field theory interacting with the classical space-time metric. One can view the classical-quantum theory as fundamental or as an effective theory useful for computing the back-reaction of quantum fields on geometry. We discuss a number of open questions from the perspective of both viewpoints.

"It's very difficult to find a black cat in a dark room, especially if there is no cat."