We were inspired to put this book together during the process of organizing the First International Conference on Quantum Error Correction at the University of Southern California (in December 2007, with a sequel in December 2011).With many of the world's foremost experts in the various branches of quantum error correction gathered together in Los Angeles, we solicited chapters on what we thought were the most important topics in the broad field. As editors, we then faced the difficult challenge of integrating material from many expert authors into one comprehensive and yet coherent volume. To achieve this feeling of coherence, we asked all our contributors to work with a single, common notation, and to work with the authors of other chapters in order to minimize overlap and maximize synchronicity. This proved hard to enforce, and while we made every effort to achieve consistency among the different chapters, this goal was surely only partly met. To the extent that the reader discovers inconsistencies, we as editors take full responsibility. The resulting book is not a textbook; for one, it doesn't include any exercises, and figures are not abundant. Moreover, it can only be a snapshot of such a rapidly evolving subject as quantum error correction. Nevertheless, we believe that the basic results in the field are now well enough established that this book, with its extensive index and list of references to the literature, will serve both as a reference for experts and as a guidebook for new researchers, for some years to come.