Advection-dominated Accretion Flows: Optically Thin Solutions

General properties of advection-dominated accretion ows are discussed. Special emphasis is given to the optically thin branch of solutions, which has very high ion and electron temperatures and is thermally stable. This solution branch has been applied to a number of low-luminosity accreting black holes. The models have resolved some puzzles and have provided a straightforward explanation of the observed spectra. The success of the models con-rms that the central objects in these low-luminosity sources are black holes. There is some indication that advection-dominated models may be relevant also for higher luminosity systems. The properties of the Low state of accreting black holes, the transition from the Low state to the High state, and the similarity of hard X-ray/-ray spectra of black hole X-ray binaries and active galactic nuclei, are explained.