In 1938, at the Institute for Advanced Study, E Hecke gave a series of lectures on his theory of correspondence between modular forms and Dirichlet series. Since then, the Hecke correspondence has remained an active feature of number theory and, indeed, it is more important today than it was in 1936 when Hecke published his original papers. This book is an amplified and up-to-date version of the former author's lectures at the University of Illinois at Urbana-Champaign, based on Hecke's notes. Providing many details omitted from Hecke's notes, it includes various new and important developments in recent years. In particular, several generalizations and analogues of the original Hecke theory are briefly described in this concise volume. Sample Chapter(s). Chapter 1: Introduction (90 KB). Contents: The Main Correspondence Theorem; A Fundamental Region; The Case λ > 2; The Case λ < 2; The Case λ = 2; Bochner's Generalization of the Main Correspondence Theorem of Hecke and Related Results; Identities Equivalent to the Functional Equation and to the Modular Relation. Readership: Graduate students interested in number theory and analysis; professional mathematicians, both experts in number theory and others interested in the Hecke theory.