Main subject categories: • Ring Theory • Polynomial Identities • Abstract Algebra[from Preface] One of the main goals of algebraists is to find large, natural classes of rings which can be analyzed in depth. An early example was Mₙ(F), the algebra of n x n matrices over a field F, for varying n and F; by the beginning of this century, the structure of Mₙ(F) was well known. Then, much important work was done on finite dimensional algebras over a field; Albert [61B] (written in 1939 and dealing exclusively with finite dimensional algebras) is still authoritative in many aspects. …There are three main aims in this book: to give some people an understandable entry into PI-theory through the first eight or nine sections of Chapter 1; to supply others with a complete account of the “state of the art”; and to point others to directions for further research. (Actually, I think further research will mostly involve the use of PI-theory in related areas.) These three aims are not always consistent, and have led to the following general guidelines:(1) Little prior knowledge is assumed (cf. the prerequisites), although it is certainly useful.(2) The point of view is not particularly modern.(3) Proofs of important results are given in detail.(4) A few areas are relegated to exercises (such as the maximal quotient ring of a semiprime PI-ring, in §1.11). The "exercises" are often sophisticated pieces of research, and hints are provided in abundance. Nevertheless, I feel little compunction in relegating them to exercises because their proofs have become so much easier in light of the new PI-theoretic techniques.