This book consists of notes for a second-year graduate course in advanced topology given by Professor George Whitehead at MIT. Presupposing a knowledge of the fundamental group and of algebraic topology as far as singular theory, it is designed to introduce the student to some of the more important concepts of homotopy theory. The book emphasizes (relative) CW-complexes, which the author believes to be the natural setting for obstruction theory, and follows the spirit of J. H. C. Whitehead's "combinatorial homotopy." Homotopy Theory will prove valuable to first- and second-year graduate students of mathematics and to mathematicians interested in this unique treatment of the subject.