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.
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