This is the first monograph on the theory of endomorphism rings of Abelian groups. The theory is a rapidly developing area of algebra and has its origin in the theory of operators of vector spaves. The text contains additional information on groups themselves, introducing new concepts, methods, and classes of groups. All the main fields of the theory of endomorphism rings of Abelian groups from early results to the most recent are covered. Neighbouring results on endomorphism rings of modules are also mentioned. The text has many pedagogical features: all the necessary definitions and formulations of assertions on Abelian groups, rings, and modules are gathered in the first two sections; summaries of results; exercises of varying difficulty in each section; lesser known facts on rings and modules are presented with proofs; and comments at the end of each chapter together with a brief historical review as well as a look at the future direction of modern research.