This book is written by a wellknown expert in classical algebraic geometry. Tyurin's research was specifically in explicit computations to vector bundles on algebraic varieties. This is the only available monograph written from his unique viewpoint. Ordinary (abelian) theta functions describe properties of moduli spaces of onedimensional vector bundles on algebraic curves. Nonabelian theta functions, which are the main topic of this book, play a similar role in the study of higherdimensional vector bundles. The book presents various aspects of the theory of nonabelian theta functions and the moduli spaces of vector bundles, including their applications to problems of quantization and to classical and quantum conformal field theories. The book is an important source of information for specialists in algebraic geometry and its applications to mathematical aspects of quantum field theory. Titles in this series are copublished with the Centre de Recherches Mathématiques. Readership Graduate students and research mathematicians interested in algebraic geometry and its applications to mathematical physics. Reviews "The book opens a beautiful and grandiose view to a fascinating part of geometry, notably symplectic and algebraic geometry, and, as usual in Andrei Tyurin's work, it has a very geometric flavor. ...The ideal reader should preferably dispose of some basic knowledge of classical algebraic geometry or theory of Riemann surfaces and of symplectic geometry, then he will benefit quite a lot from reading this book."  Zentralblatt MATH Table of Contents  Quantization procedure
 Algebraic curves = Riemann surfaces
 Nonabelian theta functions
 Symplectic geometry of moduli spaces of vector bundles
 Two versions of CQFT
 Threevalent graphs
 Analytical aspects of the theory of nonabelian theta functions
 BPUmap
 The main weapon
 Bibliography
