  We present preliminary result for the study of the renormalization group
evolution of tensor bilinears in Schr\"odinger Functional (SF) schemes for
$N_f=0$ and $N_f=2$ QCD with non-perturbatively $\mathcal{O}(a)$-improved
Wilson fermions. First $N_f=2+1$ results (proceeding in parallel with the
ongoing computation of the running quark masses [1] are also discussed. A
one-loop perturbative calculation of the discretisation effects for the
relevant step scaling functions has been carried out for both Wilson and
$\mathcal{O}(a)$-improved actions and for a large number of lattice
resolutions. We also calculate the two-loop anomalous dimension in SF schemes
for tensor currents through a scheme matching procedure with RI and
$\overline{\rm MS}$. Thanks to the SF iterative procedure the non-perturbative
running over two orders of magnitude in energy scales, as well as the
corresponding Renormalization Group Invariant operators, have been determined.
