This is an expository talk giving an overview of several results. We construct a coarse model for Teichmüller space equipped with the Teichmüller metric using a family of Gromov hyperbolic spaces called curve complexes. We then examine how geometric questions about the Teichmüller space can be translated and answered using combinatorial methods. Time permitting, we will discuss the quasi-isometric rigidity of Teichmüller space. Speaker(s): Kasra Rafi (University of Toronto)