A reference textbook that summarizes the "algebra of analysis" that includes some mention of the contemporary trends in the development of history. This text provides an up-to-date account of Banach and locally convex algebras with emphasis on general theory, representations and homology. In its description of general theory and in the examples it considers it is faithful to earlier traditions in the subject but differs from those books in being based on an approach through homological algebra. This approach leads Helemskii to consider topics not covered at this level in any other book. These topics include: complemented and uncomplemented ideal; projective and weak tensor products of banach algebras; Taylors multivariable holomorphic functional calculus; the Wedderburn decomposition of Banach algevras; a whole chapter on the homology of Banach algebras.