This book introduces a new class of non-associative algebras related to certain algebraic groups and their associated buildings. It develops a theory of these algebras, opening the first purely algebraic approach to the exceptional Moufang quadrangles. Based on their relationship to exceptional algebraic groups, quadrangular algebras belong to a series together with alternative and Jordan division algebras. Formally, the notion is derived from that of a pseudo-quadratic space over a quaternion division ring. Chapters 1-9 develop the complete classification of quadrangular algebras; in particular, every quadrangular algebra is shown to be either special, improper, regular or defective. The classification is completely elementary and (except for a few standard facts about Clifford algebras) self-contained. The book closes with a chapter on isotopes and the structure group of a quadrangular algebra.