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This monograph provides a systematic treatment of topological quantum field theories (TQFT's) in three dimensions, inspired by the discovery of the Jones polynomial of knots, the WittenChernSimons field theory, and the theory of quantum groups. The author, one of the leading experts in the subject, gives a rigorous and selfcontained exposition of new fundamental algebraic and topological concepts that emerged in this theory. The book is divided into three parts. Part I presents a construction of 3dimensional TQFT's and 2dimensional modular functors from socalled modular categories. This gives new knot and 3manifold invariants as well as linear representations of the mapping class groups of surfaces. In Part II the machinery of 6jsymbols is used to define state sum invariants of 3manifolds. Their relation to the TQFT's constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and Kauffman's skein modules. This book is accessible to graduate students in mathematics and physics with a knowledge of basic algebra and topology. It will be an indispensable source for everyone who wishes to enter the forefront of this rapidly growing and fascinating area at the borderline of mathematics and physics. Most of the results and techniques presented here appear in book form for the first time.
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