  Protoplanetary gas disks are likely to experience gravitational instabilites
(GI's) during some phase of their evolution. Density perturbations in an
unstable disk grow on a dynamic time scale into spiral arms that produce
efficient outward transfer of angular momentum and inward transfer of mass
through gravitational torques. In a cool disk with rapid enough cooling, the
spiral arms in an unstable disk form self-gravitating clumps. Whether gas giant
protoplanets can form by such a disk instability process is the primary
question addressed by this review. We discuss the wide range of calculations
undertaken by ourselves and others using various numerical techniques, and we
report preliminary results from a large multi-code collaboration. Additional
topics include -- triggering mechanisms for GI's, disk heating and cooling,
orbital survival of dense clumps, interactions of solids with GI-driven waves
and shocks, and hybrid scenarios where GI's facilitate core accretion. The
review ends with a discussion of how well disk instability and core accretion
fare in meeting observational constraints.
