Many aspects of the internal and external workings of computers can be viewed, at different levels, as a series of communication processes. Communication complexity is the mathematical theory of such communication processes. It is also often used as an abstract model of other aspects of computation. It extends Shannon's information theory, allowing two-way communication and arbitrary processes. This book surveys this mathematical theory, concentrating on the question of how much communication is necessary for any particular process. The first part of the book is devoted to the simple two-party model introduced by Yao in 1979, which is still the most widely studied model. The second part treats newer models, such as variable partition models, communication complexity of relations, and multiparty protocols, developed to deal with more complicated communication processes. Finally, applications of these models, including Turing machines, boolean circuits, computer net-works, VLSI circuits, pseudorandomness, and data structures, are treated in the third part of the book. In particular, communication arguments are used to prove lower bounds for many problems arising in these areas. This is an essential resource for graduate students and researchers in theoretical computer science, circuits, networks, VLSI, and information theory.