I review the recent progresses that have been obtained, especially through the use of high-resolution numerical simulations, on the dynamics of self-gravitating accretion discs. A coherent picture is emerging, where the disc dynamics is controlled by a small number of parameters that determine whether the disc is stable or unstable, whether the instability saturates in a self-regulated state or runs away into fragmentation, and whether the dynamics is local or global. I then apply these concepts to the case of AGN discs, discussing the implications of such evolution on the feeding of supermassive black holes. Nonfragmenting, self-gravitating discs appear to play a fundamental role in the process of formation of massive black hole seeds at high redshift ( 10–15) through direct gas collapse. On the other hand, the different cooling properties of the interstellar gas at low redshifts determine a radically different behaviour for the outskirts of the accretion discs feeding typical AGNs. Here the situation is much less clear from a theoretical point of view, and while several observational clues point to the important role of massive discs at a distance of roughly a parsec from their central black hole, their dynamics is still under debate. 1. Introduction The accretion discs surrounding the growing supermassive black holes (SMBH) in active galactic nuclei (AGN) are expected to become gravitationally unstable at a distance of ~0.01？pc from the black hole [1, 2]. Traditionally, this occurrence has been interpreted in relation to star formation: a self-gravitating disc, in this picture, would rapidly fragment and form stars [3, 4]. At the same time, it has been noted very early that the development of gravitational instability may also act as an efficient mechanism to produce torques through the effect of the resulting spiral structure and thus might be very effective in redistributing angular momentum within the disc and promote accretion [5–7]. As we shall see, the modern debate about these issues still concentrates on these two extreme cases. While we now have a much clearer understanding of the mechanism of growth and saturation of the instability in gaseous discs, and—especially though the use of high-resolution numerical simulations—we have clarified what are the main parameters regulating the disc structure and evolution, some questions are still unanswered. Are massive discs effectively truncated by star formation at the radius where they become self-gravitating, thus preventing accretion beyond these scales? Or does accretion proceed effectively
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