The study of varieties of algebraic structures, i.e. classes of algebraic structures definable by identical relations, was originated by G. Birkhoff [7] and B. H. Neumann [67] in the 1930's. A wide expansion of the ideas and methods of variety theory began in the 1950's, when the work of G. Higman, A. I. Mal'cev, B. H. Neumann, H. Neumann, A. Tarski, W. Specht was particularly influential. Since then the intensity of work in this area of algebra has remained very high, and the number of publications devoted to identities and varieties of algebraic structures is now counted in the thousands. Various aspects of the field have been systematically presented in numerous monographs and surveys — see for example [3, 12, 15, 16, 18, 64, 68, 80, 83, 84, 85, 86, 105].

As a result of this expansion, at the present moment one can speak of a number of independent but closely related algebraic theories: varieties of groups, varieties of associative algebras and polynomial identities, varieties of Lie algebras, varieties of semigroups, varieties of lattices and universal algebras, and others. We say “independent” because each of these fields has its own motivations and stimuli for development and its own natural problems. On the other hand, they are developing in close interconnection and are constantly influencing one another.

The present book is devoted to one of the newest branches of variety theory: varieties of group representations. This subject has existed for about twenty years; its foundations were laid in papers of B. I. Plotkin and his students in the late 1960's–early 1970's. There are many motivations for the study of varieties of group representations.