Publisher Summary This chapter deals with the theory of (finite dimensional) central simple algebras and the Brauer group, in particular, with division algebras. There is some question among experts as to whether this theory belongs more properly to the ring theory, field theory, cohomology theory, or algebraic K-theory. Accordingly, the chapter gives an abbreviated account of classical parts of the subject readily found elsewhere; certain key parts of the chapter (such as Brauer factor sets) draw on Jacobson's notes for the book by Jacobson and Saltman, referred to in the sequel as “JAC-SAL.” There also has been growing interest in infinite dimensional division algebras that is discussed briefly in the chapter. It develops the tools for studying the basic structure theory of central simple algebras. In view of major recent advances by Merkurjev and Suslin, the theory of central simple algebras can be presented in a remarkably clear picture. The chapter sketches this picture here and elaborates the sketch. The underlying idea is to describe central simple algebras in terms of cyclic algebras.