The Atiyah-Singer Index Formula is a deep and important result of mathematics which is known for its difficulty as well as for its applicability to a number of seemingly disparate subjects. This book is the first attempt to render this work more accessible to beginners in the field. It begins with the study of the neccessary topics in functional analysis and analysis on manifolds, and is as self-contained as possible. The third part presents the index formula and three proofs; the cobordism proof, the imbedding proof, and the heat equations proof. A section is included which surveys some of the many applications of the index formula, among them the theorem of Riemann-Roch-Hirzebruch. For this first English edition, a chapter on the applications of the Atiyah-Singer Index Formula to gauge theory has been added. This chapter also contains a discussion of Donaldson's theorem.