This is the third, substantially revised edition of this important monograph by a giant in the field of mathematics. The book is concerned with Kac-Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. Each chapter begins with a motivating discussion and ends with a collection of exercises with hints to the more challenging problems. The theory has applications in many areas of mathematics, and Lie algebras have been significant in the study of fundamental particles, including string theory, so this book should appeal to mathematical physicists, as well as mathematicians.