Computational complexity theory has developed rapidly in the past three decades. The list of surprising and fundamental results proved since 1990 alone could fill a book: these include new probabilistic definitions of classical complexity classes (IP = PSPACE and the PCP Theorems) and their implications for the field of approximation algorithms; Shor’s algorithm to factor integers using a quantum computer; an understanding of why current approaches to the famous P versus NP will not be successful; a theory of derandomization and pseudorandomness based upon computational hardness; and beautiful constructions of pseudorandom objects such as extractors and expanders. This book aims to describe such recent achievements of complexity theory in the context of the classical results. It is intended to both serve as a textbook as a reference for self-study. Thismeans it must simultaneously cater to many audiences, and it is carefully designed with that goal. Throughout the book we explain the context in which a certain notion is useful, and why things are defined in a certain way. Examples and solved exercises accompany key definitions. We assume essentially no computational background and very minimal mathematical background, which we review in Appendix A. We have also provided a web site for this book at http://www.cs.princeton.edu/theory/complexity/ with related auxiliary material. This includes web chapters on automata and computability theory,detailed teaching plans for courses based on this book, a draft of all the book’s chapters, and links to other online resources covering related topics.