In this paper, new techniques for studying the dynamics of families of surface homeomorphisms are introduced. Two dynamical deformation theories are presented --- one for surface homeomorphisms, called pruning, and another for graph endomorphisms, called kneading --- both giving conditions under which all of the dynamics in an open set can be destroyed, while leaving the dynamics unchanged elsewhere. These theories are then used to give a proof of Thurston's classification theorem for surface homeomorphisms up to isotopy.