This introduction to the theory of Teichmüller spaces will not only be used by graduate students and researchers in complex analysis and algebraic geometry, but also by theoretical physicists working with string theory. Starting with some basic facts on Riemann surfaces the authors provide a compehensive treatment of the subject and cover some very recent research topics. They have attempted to make the book as self-contained as possible and have included heuristic arguments and numerous examples.

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1 Teichmüller Space of Genus g.- 1.1 Riemann Surfaces.- 1.2 Teichmüller Space of Genus 1.- 1.3 Teichmüller Space of Genus g.- 1.4 Quasiconformal Mappings and Teichmüller Space.- 1.5 Complex Structures and Conformal Structures.- Notes.- 2 Frike Space.- 2.1 Uniformization Theorem.- 2.2 Universal Coverings.- 2.3 Möbius Transformations.- 2.4 Fuchsian Models.- 2.5 Fricke Space.- Notes.- 3 Hyperbolic Geometry and Fenchel-Nielsen Coordinates.- 3.1 Poincaré Metric and Hyperbolic Geometry.- 3.2 Fenchel-Nielsen Coordinates.- 3.3 Fricke-Klein Embedding.- 3.4 Thurston's Compactification.- Notes.- 4 Quasiconformal Mappings.- 4.1 Definitions and Elementary Properties.- 4.2 Existence Theorems on Quasiconformal Mappings.- 4.3 Dependence on Beltrami Coefficients.- 4.4 Proof of Calderón-Zygmund Theorem.- Notes.- 5 Teichmüller Spaces.- 5.1 Analytic Construction of Teichmüller Spaces.- 5.2 Teichmüller Mappings and Teichmüller's Theorerms.- 5.3 Proof of Teichmüller's Uniqueness Theorem.- Notes.- 6 Complex Analytic Theory of Teichmüller Spaces.- 6.1 Bers' Embedding.- 6.2 Invariance of Complex Structure of Teichmüller Space.- 6.3 Teichmüller Modular Groups.- 6.4 Royden's Theorems.- 6.5 Classification of Teichmüller Modular Transformations.- Notes.- 7 Weil-Petersson Metric.- 7.1 Petersson Scalar Product and Bergman Projection.- 7.2 Infinitesimal Theory of Teichmüller Spaces.- 7.3 Weil-Petersson Metric.- Notes.- 8 Fenchel-Nielsen Deformations and Weil-Petersson Metric.- 8.1 Fenchel-Nielsen Deformations.- 8.2 A Variational Formula for Geodesic Length Functions.- 8.3 Wolpert's Formula.- Notes.- Appendices.- A Classical Variations on Riemann Surfaces.- Notes.- B Compactification of the Moduli Space.- Notes.- References.- List of Symbols.