  [Abridged] Star and planet formation are the complex outcomes of
gravitational collapse and angular momentum transport mediated by protostellar
and protoplanetary disks. In this review we focus on the role of gravitational
instability in this process. We begin with a brief overview of the
observational evidence for massive disks that might be subject to gravitational
instability, and then highlight the diverse ways in which the instability
manifests itself in protostellar and protoplanetary disks: the generation of
spiral arms, small scale turbulence-like density fluctuations, and
fragmentation of the disk itself. We present the analytic theory that describes
the linear growth phase of the instability, supplemented with a survey of
numerical simulations that aim to capture the non-linear evolution. We
emphasize the role of thermodynamics and large scale infall in controlling the
outcome of the instability. Despite apparent controversies in the literature,
we show a remarkable level of agreement between analytic predictions and
numerical results. We highlight open questions related to (1) the development
of a turbulent cascade in thin disks, and (2) the role of mode-mode coupling in
setting the maximum angular momentum transport rate in thick disks.
