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.
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