  Deviations from scale invariance resulting from small perturbations of a
general two dimensional conformal field theory are studied. They are expressed
in terms of beta functions for renormalization of general couplings under local
change of scale. The beta functions for homogeneous background are given
perturbatively in terms of the data of the original conformal theory without
any specific assumptions on its nature. The renormalization of couplings to
primary operators and to first descendents is considered as well as that of
couplings of a dilatonic type which involve explicit dependence on world sheet
curvature.
