Now back in print, the revised edition of this popular study gives a systematic account of the basic results about abelian varieties. Mumford describes the analytic methods and results applicable when the ground field k is the complex field C and discusses the scheme-theoretic methods and results used to deal with inseparable isogenies when the ground field k has characteristic p. The author also provides a self-contained proof of the existence of a dual abeilan variety, reviews the structure of the ring of endormorphisms, and includes in appendices "The Theorem of Tate" and the "Mordell-Weil Thorem." This is an established work by an eminent mathematician and the only book on this subject.