  We construct a moduli space for Riemann surfaces that is universal in the
sense that it represents compact Riemann surfaces of any finite genus. This
moduli space is stratifed according to genus, and it carries a metric and a
measure that induce a Riemannian metric and a finite volume measure on each
stratum. Applications to the Plateau-Douglas problem for minimal surfaces of
varying genus and to the partition function of Bosonic string theory are
outlined. The construction starts with a universal moduli space of Abelian
varieties. This space carries a structure of an infinite dimensional locally
symmetric space which is of interest in its own right. The key to our
construction of the universal moduli space then is the Torelli map that assigns
to every Riemann surface its Jacobian and its extension to the
Satake-Baily-Borel compactifications.
