This book is the first volume of a comprehensive treatment of Teichmüller theory and its many different facets. There are 14 papers by different authors, divided into four parts. The first part is devoted to the metric and analytic theory (the Weil-Petersson metric, a harmonic map interpretation of a compactification of the Teichmüller space, the Teichmüller metric, Thurston’s asymmetric metric, decorated hyperbolic structure, Hölder distributions, the Grothendieck dessins and Teichmüller disks). Group theory aspects are studied in the second part (mapping class groups and their subgroups, deformations of Kleinian groups and a geometry of the complex of curves). Surfaces with singularities and discrete Riemann surfaces form the topic of the third part. Beautiful relations to quantum physics are discussed in the last part of the book (including quantization theories of the Teichmüller space, lamination spaces, a modular functor from quantized Teichmüller theory and a quantization of the moduli space of irreducible flat PSL(2,R) connections on a punctured surface). The first volume of the handbook already shows an extraordinarily broad spectrum of important and interesting topics related to Teichmüller theory.