  In this paper, we consider directed polymers in random environment with long
range jumps in discrete space and time. We extend to this case some techniques,
results and classifications known in the usual short range case. However, some
properties are drastically different when the underlying random walk belongs to
the domain of attraction of an $\a$-stable law. For instance, we construct
natural examples of directed polymers in random environment which experience
weak disorder in low dimension.
