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 infinite type surfaces, the aim being to study mapping class groups of infinite type surfaces via their action on this marked moduli space. We define a topology on the marked moduli space. This marked moduli space reduces to the usual Teichm"uller space for finite type surfaces. Since a big mapping class group is a topological group, a basic question is whether its action on the marked moduli space is continuous. We answer this question in the affirmative.