The first chapter of this monograph presents a survey of the theory of monotone twist maps of the annulus. First, the author covers the conservative case by presenting a short survey of AubryMather theory and Birkhoff theory, followed by some criteria for existence of periodic orbits without the areapreservation property. These are applied in the areadecreasing case, and the properties of Birkhoff attractors are discussed. A diffeomorphism of the closed annulus which is isotopic to the identity can be written as the composition of monotone twist maps. The second chapter generalizes some aspects of AubryMather theory to such maps and presents a version of the PoincaréBirkhoff theorem in which the periodic orbits have the same braid type as in the linear case. A diffeomorphism of the torus isotopic to the identity is also a composition of twist maps, and it is possible to obtain a proof of the ConleyZehnder theorem with the same kind of conclusions about the braid type, in the case of periodic orbits. This result leads to an equivariant version of the Brouwer translation theorem which permits new proofs of some results about the rotation set of diffeomorphisms of the torus. This is the English translation of a volume previously published as volume 204 in the Astérisque series. Titles in this series are copublished with Société Mathématique de France. SMF members are entitled to AMS member discounts. Readership Graduate students and research mathematicians interested in dynamical systems and geometry. Table of Contents  Presentation and comparison of the different approaches to the theory of monotone twist diffeomorphisms of the annulus
 Generating phases of the diffeomorphisms of the torus and the annulus
 Index
 Bibliography
