  We consider deformations of a conformal field theory that explicitly break
some global symmetries of the theory. If the deformed theory is still a
conformal field theory, one can exploit the constraints put by conformal
symmetry to compute broken currents anomalous dimensions. We consider several
instances of this scenario, using field theory techniques and also holographic
ones, where necessary. Field theoretical methods suffice to discuss examples of
symmetry-breaking deformations of the $O(N )$ model in $d=4-\epsilon$
dimensions. Holography is instrumental, instead, for computing current
anomalous dimensions in $\beta$-deformed superconformal field theories, and in
a class of supersymmetric RG flows at large $N$.
