Vector Bundles on Complex Projective Spaces : With an Appendix by S. I. Gelfand This expository treatment is based on a survey given by one of the authors at the Séminaire Bourbaki in November 1978 and on a subsequent course held at the University of Göttingen. It is intended to serve as an introduction to the topical question of classification of holomorphic vector bundles on complex projective spaces and can easily be read by students with a basic knowledge of analytic or algebraic geometry. Short supplementary sections describe more advanced topics further results and unsolved problems. This is a corrected third printing with an Appendix by S. I. Gelfand. The present book is the first one within the extensive literature on algebraic vector bundles to give both a self contained introduction to the basic methods and an exposition of the current state of the classification theory of algebraic vector bundles over Pn(C). (…) The reviewer thinks that readers should be grateful to the authors for presenting the first detailed self contained and systematic textbook on vector bundles over projective varieties. They have put in a lot of their own results to simplify and to systematize many proofs and to lead the reader to the current research in this field as quickly as possible. (Mathematical Reviews) (…) every section ends with historical comments further results and open questions. This brings the reader up to date and provides a guide for further work. (Bulletin of the American Mathematical Society) (…) the fundamental appendix essentially enhance this outstanding standard textbook and research monograph on vector bundles. (Mathematical Reviews)