The distribution of tunneling rates in the presence of classical chaos is derived. We use classical information about tunneling trajectories plus random matrix theory arguments about wave function overlaps. The distribution depends on the stability of a specific tunneling orbit and is not universal, though it does reduce to the universal Porter-Thomas form when the orbit is very unstable. For some situations there may be systematic deviations due to scarring of real periodic orbits. The theory is tested in a model problem and possible experimental realizations are discussed.