  Recent progress on string theory in curved spacetimes is reviewed. The string
dynamics in cosmological and black hole spacetimes is investigated.The methods
to solve the string equations of motion in curved spacetimes are described.That
is, the perturbation approach, the null approach, the $\tau$-expansion, and the
construction of global solutions.The behaviour of strings in FRW and inflatio-
nary spacetimes is obtained from the various types of string solutions. Three
different types of behaviour appear:{\bf unstable, dual} to unstable and {\bf
stable}.For the unstable strings, the energy and size grow proportional to $R$
for large scale factors $R \to \infty$. For the dual to unstable strings, the
energy and size blow up for R\to 0 as 1/R. For stable strings, the energy and
size are bounded. (In Minkowski, all solutions are of the stable type). The
self-consistent solution to the Einstein equations for string dominated
universes is reviewed. The energy-momentum tensor for a gas of strings is taken
as source and from the above behaviours the string equation of state is {\bf
derived}. The self-consistent string solution exhibits realistic matter
dominated behaviour for large times and radiation dominated behaviour for early
times. We report on the {\bf exact integrability} of the string dynamics in de
Sitter spacetime that allows to systematically find {\bf exact} string
solutions by soliton methods. {\bf Multistring solutions} are a new feature in
curved spacetimes. That is, a single world-sheet simultaneously describes many
different and independent strings. This phenomenon has no analogue in flat
spacetime and follows from the coupling of the strings with the geometry.
Finally, the string dynamics next and inside a Schwarzschild black hole is
analyzed and their physical properties discussed.
