In eukaryotic cells, degradation of most intracellular proteins is realized by proteasomes. The substrates for proteolysis are selected by the fact that the gate to the proteolytic chamber of the proteasome is usually closed, and only proteins carrying a special "label" can get into it. A polyubiquitin chain plays the role of the "label": degradation affects proteins conjugated with a ubiquitin (Ub) chain that consists at minimum of four molecules. Upon entering the proteasome channel, the polypeptide chain of the protein unfolds and stretches along it, being hydrolyzed to short peptides. Ubiquitin per se does not get into the proteasome, but, after destruction of the "labeled" molecule, it is released and labels another molecule. This process has been named "Ub-dependent protein degradation". In this review we systematize current data on the Ub-proteasome system, describe in detail proteasome structure, the ubiquitination system, and the classical ATP/Ub-dependent mechanism of protein degradation, as well as try to focus readers' attention on the existence of alternative mechanisms of proteasomal degradation and processing of proteins. Data on damages of the proteasome system that lead to the development of different diseases are given separately.