  We present a numerical study of 2D random-bond Potts ferromagnets. The model
is studied both below and above the critical value $Q_c=4$ which discriminates
between second and first-order transitions in the pure system. Two geometries
are considered, namely cylinders and square-shaped systems, and the critical
behavior is investigated through conformal invariance techniques which were
recently shown to be valid, even in the randomness-induced second-order phase
transition regime Q>4. In the cylinder geometry, connectivity transfer matrix
calculations provide a simple test to find the range of disorder amplitudes
which is characteristic of the disordered fixed point. The scaling dimensions
then follow from the exponential decay of correlations along the strip. Monte
Carlo simulations of spin systems on the other hand are generally performed on
systems of rectangular shape on the square lattice, but the data are then
perturbed by strong surface effects. The conformal mapping of a semi-infinite
system inside a square enables us to take into account boundary effects
explicitly and leads to an accurate determination of the scaling dimensions.
The techniques are applied to different values of Q in the range 3-64.
