Background charges and consistent continuous deformations of 2d gravity theories

We discuss all background charges and continuous consistent deformations of standard 2d gravity theories with scalar matter fields. The background charges and those deformations which change nontrivially the gauge symmetries are closely linked, and exist only if the target space has a covariantly constant Killing vector (a null vector for the deformations). The deformed actions provide interesting novel 2d gravity models. Some of them lead to non-critical string theories.