The theory of group representations plays an important roie in modern mathematics and its applica~ions to natural sciences. In the compulsory university curriculum it is included as a branch of algebra, dealing with representations of finite groups (see, for example, the textbook of A. I. Kostrikin ). The representation theory for compact, locally compact Abelian, and Lie groups is co- vered in graduate courses, concentrated around functional analysis. The author of the present boo~ has lectured for many years on functional analysis at Khar'kov University. He subsequently con- tinued these lectures in the form of a graduate course on the theory of group representations, in which special attention was devoted to a retrospective exposition of operator theory and harmo- nic analysis of functions from the standpoint of representation theory. In this approach it was natural to consider t only uni- tary, but also Banach representations, and t only representations of groups, but also of semigroups.