Advances on 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 : In 1987, we started a programme [1] to study the string dynamics in curved spacetime and its associated physical phenomena. This study revealed new insights and new physical phenomena with respect to string propagation in flat spacetime (and with respect to quantum fields in curved spacetime). The results are relevant both for fundamental (quantum) strings and for cosmic strings, which behave essentially in a classical way. Ap-proximative and exact solving methods have been developed. Classical and quantum string dynamics have been investigated in black hole spacetimes, cosmological backgrounds, cosmic string spacetime, gravitational wave backgrounds, supergravity backgrounds (which are necessary for fermionic strings), and near spacetime singularities. Physical phenomena like classical string instability and non oscillatory motion in time, quantum particle transmutation, string scattering, string stretching, have been found.