This volume is the first in a projected series devoted to the mathematical and philosophical works of the late Claude Chevalley. It covers the main contributions by the author to the theory of spinors. Since its appearance in 1954, "The Algebraic Theory of Spinors" has been a much sought after reference. It presents the whole story of one subject in a concise and especially clear manner. The reprint of the book is supplemented by a series of lectures on Clifford Algebras given by the author in Japan at about the same time. Also included is a postface by J.-P. Bourguignon describing the many uses of spinors in differential geometry developed by mathematical physicists from the 1970s to the present day. An insightful review of "Spinors" by J. Dieudonne is also made available to the reader in this new edition.