The problem of how to parametrize the variations of complex structures on a fixed base surface originated with G. F. Bernhard Riemann. This problem has spurred extensive investigations, and progress has been considerable in the areas of the theory of Riemann surfaces, algebraic geometry, and differential geometry.
In this book, we present the theory of Teichmiiller spaces which give a parametrization of all the complex structures on a given surface. This subject lies in the intersection of many important areas of mathematics. These include complex manifolds, Fuchsian groups, Kleinian groups, Lie groups, automorphic forms, complex analysis, algebraic geometry, differential geometry, topology in two and three dimensions, differential equations, complex dynamics, and ergodic theory. Recently, the theory of Teichmiiller spaces has begun playing important roles in the string theory.
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