  We construct and discuss all background charges and continuous consistent
deformations of standard $2d$ gravity theories with scalar matter fields. It
turns out that the background charges and those deformations which change
nontrivially both the form of the action and of its gauge symmetries are
closely linked and exist only if the target space has at least one special
(`covariantly constant') Killing vector which must be a null vector in the case
of the deformations. The deformed actions provide interesting novel $2d$
gravity models. We argue that some of them lead to non-critical string
theories.
