The Teichmuller space $T(X)$ is the space of marked conformal structures on a given quasi conformal surface $X$. This volume uses quasi conformal mapping to give a unified and up-to-date treatment of $T(X)$. Emphasis is placed on parts of the theory applicable to non compact surfaces and to surfaces possibly of infinite analytic type. The book provides a treatment of deformations of complex structures on infinite Riemann surfaces and gives background for further research in many areas. These include applications to fractal geometry, to three-dimensional manifolds through its relationship to Kleinian groups, and to one-dimensional dynamics through its relationship to quasi symmetric mappings. Many research problems in the application of function theory to geometry and dynamics are suggested.