The current state of research on muon catalysis is presented. Means of possibly using this phenomenon for the production of nuclear energy and neutrons are discussed, as is the role of muon catalysis in a number of other alternative methods of nuclear breeding. Basic unsolved problems of muon catalysis and its use for solving various problems in nuclear and atomic physics, quantum electrodynamics, computational physics, etc., are analyzed. An extensive bibliography is provided of basic work carried out since the discovery of the phenomenon of muon catalysis.