  In this paper we study the topology of the space of Riemann surfaces in a
simply connected space X, S_{g,n} (X, \gamma). This is the space consisting of
triples, (F_{g,n}, \phi, f), where F_{g,n} is a Riemann surface of genus g and
n-boundary components, \phi is a parameterization of the boundary, and f :
F_{g,n} \to X is a continuous map that satisfies a boundary condition \gamma.
We prove three theorems about these spaces. Our main theorem is the
identification of the stable homology type of the space S_{\infty, n}(X;
\gamma), defined to be the limit as the genus g gets large, of the spaces
S_{g,n} (X; \gamma). Our result about this stable topology is a parameterized
version of the theorem of Madsen and Weiss proving a generalization of the
Mumford conjecture on the stable cohomology of mapping class groups. Our second
result describes a stable range in which the homology of S_{g,n} (X; \gamma) is
isomorphic to the stable homology. Finally we prove a stability theorem about
the homology of mapping class groups with certain families of twisted
coefficients. The second and third theorems are generalizations of stability
theorems of Harer and Ivanov.
