From author: ''The exposition in the following pages is an elaboration and expansion of lectures I gave to second year mathematics graduate students at Wesleyan University during the academic years 1970-71 and 1971-72. The aim of the exposition is to provide a compact introduction to the theory of Banach algebras that not only acquaints the reader with fundamental portions of the abstract theory but also illustrates the usefulness of Banach algebras in the study of harmonic analysis and function algebras and gives the reader the basic tools necessary for further work in these areas. The first half of the book is devoted primarily to the general theory of Banach algebras, while in the second half the emphasis is on various more specialized topics related to harmonic analysis and function algebras - among which are: Wiener's Tauberian Theorem, the problem of spectral synthesis, the Bishop, Choquet, and 5ilov boundaries, representing measures, Werner's Maximality Theorem, the Commutative Gel'fand-Naimark Theorem, Plancherel's Theorem, the Pontryagin Duality Theorem, almost periodic functions, and the Bohr compactification.An intelligent reading of the book presupposes the usual mathematical equipment possessed by second year mathematics graduate students with regard to topology, algebra, and real and complex analysis, as well as a reasonably good knowledge of basic functional analysis...''