This is a book in pure mathematics dealing with homotopy theory, one of the main branches of algebraic topology. The principal topics are as follows: Basic Homotopy, H-spaces and co-H-spaces, fibrations and cofibrations, exact sequences of homotopy sets, actions, and coactions, homotopy pushouts and pullbacks, classical theorems, including those of Serre, Hurewicz, Blakers-Massey, and Whitehead, homotopy Sets, homotopy and homology decompositions of spaces and maps, and obstruction theory. The underlying theme of the entire book is the Eckmann-Hilton duality theory. The book can be used as a text for the second semester of an advanced ungraduate or graduate algebraic topology course.