On directed polymers in a random environment

We consider random walks whose laws are perturbed in an irregular way by a second random mechanism, the so-called random environment. The perturbation acts in such a way that visits to certain randomly chosen points are favoured or disfavoured. Several models have been proposed in the mathematical physics literature, sharing the common feature that only very few facts are known at a rigorous mathematical level. We present here some of these models and some problems connected with them