  I review recent progresses in the dynamics and the evolution of
self-gravitating accretion discs. Accretion discs are a fundamental component
of several astrophysical systems on very diverse scales, and can be found
around supermassive black holes in Active Galactic Nuclei (AGN), and also in
our Galaxy around stellar mass compact objects and around young stars.
Notwithstanding the specific differences arising from such diversity in
physical extent, all these systems share a common feature where a central
object is fed from the accretion disc, due to the effect of turbulence and disc
instabilities, which are able to remove the angular momentum from the gas and
allow its accretion. In recent years, it has become increasingly apparent that
the gravitational field produced by the disc itself (the disc's self-gravity)
is an important ingredient in the models, especially in the context of
protostellar discs and of AGN discs. Indeed, it appears that in many cases (and
especially in the colder outer parts of the disc) the development of
gravitational instabilities can be one of the main agents in the redistribution
of angular momentum. In some cases, the instability can be strong enough to
lead to the formation of gravitationally bound clumps within the disc, and thus
to determine the disc fragmentation. As a result, progress in our understanding
of the dynamics of self-gravitating discs is essential to understand the
processes that lead to the feeding of both young stars and of supermassive
black holes in AGN. At the same time, understanding the fragmentation
conditions is important to determine under which conditions AGN discs would
fragment and form stars and whether protostellar discs might form giant gaseous
planets through disc fragmentation.
