  The complete set of solutions of two dimensional classical string theory are
constructed for any curved spacetime. They describe folded strings moving in
curved spacetime. Surprizing stringy behavior becomes evident at singularities
such as black holes.The solutions are given in the form ofa map from the world
sheet to target spacetime, where the world sheet has to be divided into lattice
-like patches corresponding to different maps. A recursion relation analogous
to a transfer matrix connects these maps into a single continuous map. This
``transfer matrix'' encodes the properties of the world sheet lattice as well
as the geometry of spacetime. The solutions are completely classified by their
behavior in the asymptotically flat region of spacetime where they reduce, as
boundary conditions, to the folded string solutions that have been known for 19
years.
