A fundamental element of the study of 3manifolds is Thurston's remarkable geometrization conjecture, which states that the interior of every compact 3manifold has a canonical decomposition into pieces that have geometric structures. In most cases, these structures are complete metrics of constant negative curvature, that is to say, they are hyperbolic manifolds. The conjecture has been proved in some important cases, such as Haken manifolds and certain types of fibered manifolds. The influence of Thurston's hyperbolization theorem on the geometry and topology of 3manifolds has been tremendous. This book presents a complete proof of the hyperbolization theorem for 3manifolds that fiber over the circle, following the plan of Thurston's original (unpublished) proof, though the double limit theorem is dealt with in a different way. The book is suitable for graduate students with a background in modern techniques of lowdimensional topology and will also be of interest to researchers in geometry and topology. This is the English translation of a volume originally published in 1996 by the Société Mathématique de France. 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 lowdimensional topology and geometry. Reviews From a review of the French edition: "The book is very well written ... completely selfcontained ..."  Mathematical Reviews Table of Contents  Teichmüller spaces and Kleinian groups
 Real trees and degenerations of hyperbolic structures
 Geodesic laminations and real trees
 Geodesic laminations and the Gromov topology
 The double limit theorem
 The hyperbolization theorem for fibered manifolds
 Sullivan's theorem
 Actions of surface groups on real trees
 Two examples of hyperbolic manifolds that fiber over the circle
 Geodesic laminations
 Bibliography
 Index
