  We study the effect of interfacial phenomena in two-dimensional perfect and
random (or disordered) $q$-state Potts models with continuous phase
transitions, using, mainly, Monte Carlo techniques. In particular, for the
total interfacial adsorption, the critical behavior, including corrections to
scaling, are analyzed. The role of randomness is scrutinized. Results are
discussed applying scaling arguments and invoking findings for bulk critical
properties. In all studied cases, i.e., $q = 3$, $4$, and $q = 8$, the spread
of the interfacial adsorption profiles is observed to increase linearly with
the lattice size at the bulk transition point.
