  We calculate the all-loop anomalous dimensions of current operators in
$\lambda$-deformed $\sigma$-models. For the isotropic integrable deformation
and for a semi-simple group $G$ we compute the anomalous dimensions using two
different methods. In the first we use the all-loop effective action and in the
second we employ perturbation theory along with the Callan-Symanzik equation
and in conjunction with a duality-type symmetry shared by these models.
Furthermore, using CFT techniques we compute the all-loop anomalous dimensions
of bilinear currents for the isotropic deformation case and a general $G$.
Finally we work out the cases of anisotropic $SU(2)$ and the two coupling,
corresponding to the symmetric coset $G/H$ and a subgroup $H$, splitting of a
group $G$.
