Advances in String Theory in Curved Space Times

String Quantum Gravity is motivated and introduced. Advances in the study of the classical and quantum string dynamics in curved spacetime are reported: 1-New Classes of Exact Multistring solutions in curved spacetimes. 2-Mass spectrum of Strings in Curved Spacetimes. 3-The effect of a Cosmological Constant and of Spacial Curvature on Classical and Quantum Strings. 4-Classical splitting of Fundamental Strings. 5-The General String Evolution in constant Curvature Spacetimes. 6-The Conformal Invariance Effects.