  Tensor currents are the only quark bilinear operators lacking a
non-perturbative determination of their renormalisation group (RG) running
between hadronic and electroweak scales. We develop the setup to carry out the
computation in lattice QCD via standard recursive finite-size scaling
techniques, and provide results for the RG running of tensor currents in $N_f =
0$ and $N_f = 2$ QCD in the continuum for various Schr\"odinger Functional
schemes. The matching factors between bare and renormalisation group invariant
currents are also determined for a range of values of the lattice spacing
relevant for large-volume simulations, thus enabling a fully non-perturbative
renormalization of physical amplitudes mediated by tensor currents.
