  It is quite likely that self-gravity will play an important role in the
evolution of accretion discs, in particular those around young stars, and those
around supermassive black holes. We summarise, here, our current understanding
of the evolution of such discs, focussing more on discs in young stellar
system, than on discs in active galactic nuclei. We consider the conditions
under which such discs may fragment to form bound objects, and when they might,
instead, be expected to settle into a quasi-steady, self-regulated state. We
also discuss how this understanding may depend on the mass of the disc relative
to the mass of the central object, and how it might depend on the presence of
external irradiation. Additionally, we consider whether or not fragmentation
might be stochastic, where we might expect it to occur in an actual
protostellar disc, and if there is any evidence for fragmentation actually
playing a role in the formation of planetary-mass bodies. Although there are
still a number of outstanding issue, such as the convergence of simulations of
self-gravitating discs, whether or not there is more than one mode of
fragmentation, and quite what role self-gravitating discs may play in the
planet formation process, our general understanding of these systems seems
quite robust.
