Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. It applies ideas from theoretical physics to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space. This book is the first systematic exposition of Homotopy Quantum Field Theory. It starts with a formal definition of an HQFT and provides examples of HQFTs in all dimensions. The main body of the text is focused on \(2\)dimensional and \(3\)dimensional HQFTs. A study of these HQFTs leads to new algebraic objects: crossed Frobenius groupalgebras, crossed ribbon groupcategories, and Hopf groupcoalgebras. These notions and their connections with HQFTs are discussed in detail. The text ends with several appendices including an outline of recent developments and a list of open problems. Three appendices by M. Müger and A. Virelizier summarize their work in this area. The book is addressed to mathematicians, theoretical physicists, and graduate students interested in topological aspects of quantum field theory. The exposition is selfcontained and well suited for a onesemester graduate course. Prerequisites include only basics of algebra and topology. 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 topological aspects of quantum field theory. Table of Contents  Generalities on HQFTs
 Groupalgebras
 Twodimensional HQFTs
 Biangular algebras and lattice HQFTs
 Enumeration problems in dimension two
 Crossed \(G\)categories and invariants of links
 Modular \(G\)categories and HQFTs
 Miscellaneous algebra
 Appendix 1. Relative HQFTs
 Appendix 2. State sum invariants of 3dimensional \(G\)manifolds
 Appendix 3. Recent work on HQFTs
 Appendix 4. Open problems
 Appendix 5. On the structure of braided crossed \(G\)categories
 Appendix 6. Algebraic properties of Hopf \(G\)coalgebras
 Appendix 7. Invariants of 3dimensional \(G\)manifolds from Hopf coalgebras
 Bibliography
 Index
