This volume is the fifth in a series dedicated to Teichmuller theory in a broad sense, including the study of various deformation spaces and of mapping class group actions. It is divided into four parts: Part A: The metric and the analytic theory Part B: The group theory Part C: Representation theory and generalized structures Part D: Sources The topics that are covered include identities for the hyperbolic geodesic length spectrum, Thurston's metric, the cohomology of moduli space and mapping class groups, the Johnson homomorphisms, Higgs bundles, dynamics on character varieties, and many others. Besides surveying important parts of the theory, several chapters contain conjectures and open problems. The last part contains two fundamental papers by Teichmuller, translated into English and accompanied by mathematical commentaries. The papers, like those of the other volumes in this collection, are written by experts who have a broad view on the subject. Although the papers have an expository character (which fits with the purpose of the handbook), some of them also contain original and new material.