During the 1980s, profound connections were discovered relating modern algebraic geometry and algebraic \(K\)-theory to arithmetic problems. The term “arithmetic algebraic geometry” was coined during that period and is now used to denote an entire branch of modern number theory. These same developments in algebraic geometry and \(K\)-theory greatly influenced research on the arithmetic of fields in general, and the algebraic theory of quadratic forms and the theory of finite-dimensional division algebras in particular. This book contains papers presented at an AMS Summer Research Institute held in July 1992 at the University of California, Santa Barbara. The purpose of the conference was to provide a broad overview of the tools from algebraic geometry and \(K\)-theory that have proved to be the most powerful in solving problems in the theory of quadratic forms and division algebras. In addition, the conference provided a venue for exposition of recent research. A substantial portion of the lectures of the major conference speakers—Colliot-Thélène, Merkurjev, Raskind, Saltman, Suslin, Swan—are reproduced in the expository articles in this book.