An account of one of the main directions of algebraic topology, this book focuses on the Sullivan conjecture and its generalizations and applications. It gathers work on the theory of modules over the Steenrod algebra, including ideas on the Segal conjecture, work from the late 1970s by Adams and Wilkerson, and topics in algebraic group representation theory. This course-tested book is intended to be of use as a reference for algebraic topologists and includes foundational material that is of relevance for graduate study.