The central topic of this book is the theorem of Hurwitz and Radon concerning composition formulas for sums of squares, first proved in the 1920's. Techniques from algebra and topology are used to generalize that theorem in several directions. The text includes worked examples and many exercises, which develop still more variations of the central topic. The main audience is people who have had some graduate courses in abstract algebra, but many sections of the book are accessible to anyone with some training in linear algebra. Several major topics in this book will be of interest to students of topology and geometry. The author has attempted to make the presen-tation as clear and as elementary as possible.