This book provides a detailed introduction to linear wave equations on Lorentzian manifolds (for vectorbundle valued fields). After a collection of preliminary material in the first chapter, one finds in the second chapter the construction of local fundamental solutions together with their Hadamard expansion. The third chapter establishes the existence and uniqueness of global fundamental solutions on globally hyperbolic spacetimes and discusses Green's operators and wellposedness of the Cauchy problem. The last chapter is devoted to field quantization in the sense of algebraic quantum field theory. The necessary basics on \(C^*\)algebras and CCRrepresentations are developed in full detail. The text provides a selfcontained introduction to these topics addressed to graduate students in mathematics and physics. At the same time, it is intended as a reference for researchers in global analysis, general relativity, and quantum field theory. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in mathematical physics. Table of Contents  Preliminaries
 The local theory
 The global theory
 Quantization
 Appendix. Background material
 Bibliography
 Figures
 Symbols
 Index
