Asymptotic geometry is the study of metric spaces from a large scale point of view, where the local geometry does not come into play. An important class of model spaces are the hyperbolic spaces (in the sense of Gromov), for which the asymptotic geometry is nicely encoded in the boundary at infinity. In the first part of this book, in analogy with the concepts of classical hyperbolic geometry, the authors provide a systematic account of the basic theory of Gromov hyperbolic spaces. These spaces have been studied extensively in the last twenty years and have found applications in group theory, geometric topology, Kleinian groups, as well as dynamics and rigidity theory. In the second part of the book, various aspects of the asymptotic geometry of arbitrary metric spaces are considered. It turns out that the boundary at infinity approach is not appropriate in the general case, but dimension theory proves useful for finding interesting results and applications. The text leads concisely to some central aspects of the theory. Each chapter concludes with a separate section containing supplementary results and bibliographical notes. Here the theory is also illustrated with numerous examples as well as relations to the neighboring fields of comparison geometry and geometric group theory. The book is based on lectures the authors presented at the Steklov Institute in St. Petersburg and the University of Zürich. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and researchers working in geometry, topology, and geometric group theory. Table of Contents  Hyperbolic geodesic spaces
 The boundary at infinity
 Busemann functions on hyperbolic spaces
 Morphisms of hyperbolic spaces
 QuasiMöbius and quasisymmetric maps
 Hyperbolic approximation of metric spaces
 Extension theorems
 Embedding theorems
 Basics of dimension theory
 Asymptotic dimension
 Linearly controlled metric dimension: Basic properties
 Linearly controlled metric dimension: Applications
 Hyperbolic dimension
 Hyperbolic rank and subexponential corank
 Appendix. Models of the hyperbolic space \(H^n\)
 Bibliography
 Index
