Since its discovery in 1997 by Maldacena, AdS/CFT correspondence has become one of the prime subjects of interest in string theory, as well as one of the main meeting points between theoretical physics and mathematics. On the physical side it provides a duality between a theory of quantum gravity and a field theory. The mathematical counterpart is the relation between Einstein metrics and their conformal boundaries. The correspondence has been intensively studied, and a lot of progress emerged from the confrontation of viewpoints between mathematics and physics. Written by leading experts and directed at research mathematicians and theoretical physicists as well as graduate students, this volume gives an overview of this important area both in theoretical physics and in mathematics. It contains survey articles giving a broad overview of the subject and of the main questions, as well as more specialized articles providing new insight both on the Riemannian side and on the Lorentzian side of the theory. Table of Contents: Preface v Michael T. Anderson Geometric aspects of the AdS/CFT correspondence 1 Jan de Boer, Liat Maoz and Asad Naqvi Some aspects of the AdS/CFT correspondence 33 C.Robin Graham and Kengo Hirachi The ambient obstruction tensor and $Q$-curvature 59 Ioannis Papadimitriou and Kostas Skenderis AdS/CFT correspondence and geometry 73 Marc Herzlich Mass formulae for asymptotically hyperbolic manifolds 103 Sergey N. Solodukhin Reconstructing Minkowski space-time 123 Michael T. Anderson, Piotr T. Chru\´sciel and Erwann Delay Non-trivial, static, geodesically complete space-times with a negative cosmological constant II. $n\ge 5$ 165 Charles Frances The conformal boundary of anti-de Sitter space-times 205 Jerome P. Gauntlett, Dario Martelli, James Sparks and Daniel Waldram Supersymmetric AdS backgrounds in string and M-theory 217