  In this lecture I summarize recent developments on strings propagating in
curved spacetime. Exact conformal field theories that describe gravitational
backgrounds such as black holes and more intricate gravitational singularities
have been discovered and investigated at the classical and quantum level. These
models are described by gauged Wess-Zumino-Witten models, or equivalently
current algebra G/H coset models based on non-compact groups, with a single
time coordinate. The classification of such models for all dimensions is
complete. Furthermore the heterotic superstrings in curved spacetime based on
non-compact groups have also been constructed. For many of the $d\le 4$ models
the gravitational geometry described by a sigma model has been determined. Some
general results outlined here include a global analysis of the geometry and the
exact classical geodesics for any G/H model. Moreover, in the quantized theory,
the conformally exact metric and dilaton are obtained for all orders in an
expansion of $k$ (the central extension). All such models have large-small (or
mirror) duality properties which we reformulate as an inversion in group space.
To illustrate model building techniques a specific 4-dimensional heterotic
string in curved spacetime is presented. Finally the methods for investigating
the quantum theory are outlined. The construction and analysis of these models
at the classical and quantum level involve some aspects of noncompact groups
which are not yet sufficiently well understood. Some of the open problems in
the physics and mathematics areas are outlined.
