  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.
