Curved Spacetime Geometry for Strings and Affine Non-compact Algebras

Based on lectures delivered at (1) the AMS meeting at USC, Nov. 1992 (2) Conference on Quantum Aspects of Black Holes, ITP, UC- Santa %Barbara, %%June 1993. (3) 25th Summer Institute, Ecole Normale Superieure, % Paris, Aug. 1992. To appear in "Interface Between Mathematics and Physics", Ed. S.-T. Yau. 1--Itroduction to String Theory in Curved Spacetime 2--G/H Coset Conformal Field Theory and String Theory 3--Time, Space and Classification of Non-Compact Cosets 4--Heterotic Strings in Curved Spacetime as gauged WZW Models 5--The Spacetime Manifold and the Geometry 6--Examples in 2D, 3D and 4D