We show that a large class of dissipative systems can be brought to a canonical form by introducing complex co-ordinates in phase space and a complex-valued hamiltonian. A naive canonical quantization of these systems lead to non-hermitean hamiltonian operators. The excited states are unstable and decay to the ground state. We also compute the tunneling amplitude across a potential barrier by solving the dissipative version of the Schrödinger equation. We then generalize the formalism to cases where the configuration space is a curved Riemannian manifold.