Accretion disks play a central role in essentially all compact astrophysical systems, from active galactic nuclei on galactic scales, to X-ray binaries, microquasars and gamma-ray bursts on stellar scales. They represent the accumulation of angular momentum in the attraction of matter from the host environment – the ionized medium around supermassive black holes provided by stellar winds and/or tidally disrupted stars, Roche lobe overflow from a companion star in compact binaries, and fall back matter from the envelope of a collapsed star in GRBs. Their fluid dynamical properties are key to their radiative signatures, stability, wave modes and outflows. To leading order, an accretion disk assumes a largely Keplerian motion, subject to inflow of matter from larger radii and a consequent outflow of angular momentum. Dissipation of their rotational energy and the resulting radial motion are mediated by some form of macroscopic viscosity, as will be discussed below. A fraction of the binding energy may be released as disk winds, further complicating the system. Moreover, for certain configurations gravitational-wave emissions may become appreciable, opening another window into the physics of black holes and their accretion disks.

On large scales, accretion disks can be observed directly, such as the disk in NGC 4256 with its rotational motion measured by Doppler shifts of its maser emissions. X-ray spectroscopy and other techniques can be exploited to probe rotational motion of accreted matter around supermassive black holes and stellar mass compact objects down to the innermost regions, on angular scales much smaller than directly accessible by current instruments.