This book is intended to be a self-contained introduction to all the set theory needed by most mathematicians. It is self-contained in the sense that no prior knowledge of set theory is logically assumed; but the reader should have a working ability in the elements of the subject—the topics covered in Chap. 1. The usual elementary definitions and facts about sets, relations, functions, unions, and so on, are covered in Chap. 1. In Chap. 2 ordinal numbers are studied; the main topics are definitions by transfinite recursions and the rudiments of ordinal arithmetic. Chapter 3 is devoted to a brief study of the axiom of choice; several equivalent forms of this axiom are given, and some mathematical applications are discussed. The heart of the book is Chap. 4, on cardinal numbers. Besides elementary facts about addition, multiplication, and exponentiation of cardinals, some more advanced cardinal arithmetic is discussed; in particular, regular and singular cardinals are treated at some length.