The study of derived categories is a subject that attracts increasingly many mathematicians from various fields of mathematics, including abstract algebra, algebraic geometry, representation theory, and mathematical physics. The concept of the derived category of sheaves was invented by Grothendieck and Verdier in the 1960s as a tool to express important results in algebraic geometry such as the duality theorem. In the 1970s, Beilinson, Gelfand, and Gelfand discovered that a derived category of an algebraic variety may be equivalent to that of a finitedimensional noncommutative algebra, and Mukai found that there are nonisomorphic algebraic varieties that have equivalent derived categories. In this way, the derived category provides a new concept that has many incarnations. In the 1990s, Bondal and Orlov uncovered an unexpected parallelism between the derived categories and the birational geometry. Kontsevich's homological mirror symmetry provided further motivation for the study of derived categories. This book contains the proceedings of a conference held at the University of Tokyo in January 2011 on the current status of the research on derived categories related to algebraic geometry. Most articles are survey papers on this rapidly developing field. The book is suitable for mathematicians who want to enter this exciting field. Some basic knowledge of algebraic geometry is assumed. 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 derived categories in algebraic geometry. Table of Contents  M. Bernardara and M. Bolognesi  Categorical representability and intermediate Jacobians of Fano threefolds
 A. Canonaco and P. Stellari  FourierMukai functors: A survey
 S. Cautis  Flops and about: A guide
 A. Ishii and K. Ueda  A note on derived categories of Fermat varieties
 D. Kaledin  Homology of infinite loop spaces
 B. Keller  Cluster algebras and derived categories
 I. Mori  Some derived equivalences between noncommutative schemes and algebras
 A. Polishchuk  Lagrangianinvariant sheaves and functors for abelian varieties
 M. Popa  Generic vanishing filtrations and perverse objects in derived categories of coherent sheaves
 C. Schnell  The fundamental group is not a derived invariant
 Y. Toda  Introduction and open problems of DonaldsonThomas theory
 M. Van den Bergh  Notes on formal deformations of abelian categories
 List of contributors
