  Progress on string theory in curved spacetimes since 1992 are reviewed. After
a short introduction on strings in Minkowski and curved spacetimes, we focus on
strings in cosmological spacetimes. The classical behaviour of strings in FRW
and inflationary spacetimes is now understood in a large extent from various
types of explicit string solutions. Three different types of behaviour appear
in cosmological spacetimes: {\bf unstable, dual} to unstable and {\bf stable}.
For the unstable strings, the energy and size grow for large scale factors $R
\to \infty$, proportional to $R$. 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
proper size are bounded. (In Minkowski spacetime, all string solutions are of
the stable type).Recent progress on self-consistent solutions to the Einstein
equations for string dominated universes is reviewed. The energy-momentum
tensor for a gas of strings is then considered as source of the spacetime
geometry and from the above string behaviours the string equation of state is
determined. The self-consistent string solution exhibits the realistic matter
dominated behaviour $R \sim (X^0)^{2/3}\; $ for large times and the radiation
dominated behaviour $R \sim (X^0)^{1/2}\;$ for early times. Finally, we report
on the exact integrability of the string equations plus the constraints in de
Sitter spacetime that allows to systematically find {\bf exact} string
solutions by soliton methods and the multistring solutions. {\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 to the coupling
of the strings with the geometry.
