We introduce the moduli space of marked, complete, Nielsen-convex hyperbolic structures on a surface of negative, but not necessarily finite, Euler characteristic. The emphasis is on the case in which the surface is of infinite type, the aim being to study the mapping class group of such a surface via its action on this marked moduli space. In this dissertation, we lay the foundations for the study of the marked moduli space and this action. We define a topology on the marked moduli space and describe it in various ways, each reminiscent of the Teichmüller space. We prove that it reduces to the usual Teichmüller space in case the surface is of finite type. We prove that the above natural action of the mapping class group on the marked moduli space by change of marking is continuous. We show that the marked moduli space is contractible. The proof is uniform for all surfaces. We also investigate the geometry of the marked moduli space. We show that for certain classes of surfaces, the marked moduli space does not admit any mapping class group invariant metrics.