This book provides background for applications of Teichmuller theory to dynamical systems and, in particular, to iteration of rational maps and conformal dynamics, to Kleinian groups and three dymensional manifolds, to Fuchsian groups and Riemann surfaces, and to one-dimensional dynamics. The focus is on new developments in the theory in both finite and infinite dimensional cases. An exposition of the main theorems is given regardless of dimensionality by using techniques that apply generally. The hope is to provide background for more applications.