This book, "Vector Bundles on Complex Projective Spaces," edited by J. Coates and S. Helgason, is a comprehensive introduction to the classification of holomorphic vector bundles over projective algebraic manifolds, primarily focusing on the case where the manifold is $\mathbb{P}_n$. The authors, Christian Okonek, Michael Schneider, and Heinz Spindler, aim to provide a concrete and accessible treatment, using mostly the language of analytic geometry. The book is divided into two chapters, each containing several sections that cover various aspects of vector bundles, including basic definitions, the splitting of vector bundles, uniform bundles, examples of indecomposable bundles, holomorphic 2-bundles, and the existence of holomorphic structures on topological bundles. The second chapter delves into stability and moduli spaces, discussing stable bundles, the splitting behavior of stable bundles, monads, and the moduli of stable 2-bundles. The book includes historical remarks, further results, and open problems at the end of each section, making it a valuable resource for students and researchers in the field of algebraic and analytic geometry.This book, "Vector Bundles on Complex Projective Spaces," edited by J. Coates and S. Helgason, is a comprehensive introduction to the classification of holomorphic vector bundles over projective algebraic manifolds, primarily focusing on the case where the manifold is $\mathbb{P}_n$. The authors, Christian Okonek, Michael Schneider, and Heinz Spindler, aim to provide a concrete and accessible treatment, using mostly the language of analytic geometry. The book is divided into two chapters, each containing several sections that cover various aspects of vector bundles, including basic definitions, the splitting of vector bundles, uniform bundles, examples of indecomposable bundles, holomorphic 2-bundles, and the existence of holomorphic structures on topological bundles. The second chapter delves into stability and moduli spaces, discussing stable bundles, the splitting behavior of stable bundles, monads, and the moduli of stable 2-bundles. The book includes historical remarks, further results, and open problems at the end of each section, making it a valuable resource for students and researchers in the field of algebraic and analytic geometry.