Determines local Euler factors on classical groups in an explicitly rational form, Euler products and Eisenstein series on a unitary group of an arbitrary signature, and a class number formula for a totally definite hermitian form. Though the results are new and previously unpublished, expository material is included to make the treatment suitable as an introduction to the theory of automorphic forms of several variables, particularly Hecke theory, for readers familiar with algebraic number theory. Shimura (Princeton U.) delivered the lectures at Texas Christian University in May 1996. Annotation c. by Book News, Inc., Portland, Or.