In a protoplanetary disk that is sufficiently cold and massive, gravitational instabilities (GIs) will lead to the development of dense spiral waves on a dynamic time scale. For sufficiently short cooling times, comparable to about half a rotation period, an unstable disk will fragment into dense clumps that could be the precursors of gas giant protoplanets. At moderate cooling rates, the strong spiral waves which permeate the disk do not fragment, but nevertheless generate significant mass and angular momentum transport. I will review recent research on GIs with an emphasis on several critical questions: Do GIs cause planets to form? How fast do they transport mass? When do they occur? How do they affect the solids in the disk? The physical processes that are central to answering these questions are radiative and possibly convective cooling, irradiation of the disk, and gas-solid interactions. I conclude that, while it is unlikely that gas giant planets are formed directly by disk instability, GIs may substantially accelerate both planetesimal formation and core accretion.