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We give an overview of the state of the art of the analysis of disorderedmodels of pinning on a defect line. This class of models includes a number ofwell known and much studied systems (like polymer pinning on a defect line,wetting of interfaces on a disordered substrate and the Poland-Scheraga modelof DNA denaturation). A remarkable aspect is that, in absence of disorder, allthe models in this class are exactly solvable and they display alocalization-delocalization transition that one understands in full detail.Moreover the behavior of such systems near criticality is controlled by aparameter and one observes, by tuning the parameter, the full spectrum ofcritical behaviors, ranging from first order to infinite order transitions.This is therefore an ideal set-up in which to address the question of theeffect of disorder on the phase transition,notably on critical properties. Wewill review recent results that show that the physical prediction that goesunder the name of Harris criterion is indeed fully correct for pinning models.Beyond summarizing the results, we will sketch most of the arguments of proof.

Renewal sequences, disordered potentials, and pinning phenomena