Random polymers and delocalization transitions

In these proceedings, we first summarize some general properties of phase transitions in the presence of quenched disorder, with emphasis on the following points: the need to distinguish typical and averaged correlations, the possible existence of two correlation length exponents $\nu$, the general bound $\nu_{FS} \geq 2/d$, the lack of self-averaging of thermodynamic observables at criticality, the scaling properties of the distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples of size $L$. We then review our recent works on the critical properties of various delocalization transitions involving random polymers, namely (i) the bidimensional wetting (ii) the Poland-Scheraga model of DNA denaturation (iii) the depinning transition of the selective interface model (iv) the freezing transition of the directed polymer in a random medium.