Directed polymers in a random environment is a model of statistical mechanics introduced in the 80s. Given a set if independent, identically random variable n;x indexed by N  Zd, and a parameter > 0, it is de ned as the measure on the set of nearest neighbor path of length N, (Sn)N n=0 which to each path gives a weight proportional to QN i=1(1 + n;Sm).
The aim of this talk is to discuss the existence of a continuum scaling limit for this model if the parameter is sent to zero when N tends to in nity. We will rst introduce the topic to the audience by reviewing a result of Alberts, Khanin and Quastel in 2014 which establishes the existence of such a scaling limit in dimension 1 under the assumption that the environment has a nite second moment. Afterwards we will present our main results which:

1. Establishes the existence of another family of continuum model based on -stable noise rather than Gaussian white noise,
2. Show that this continuum model can be obtained as the scaling limit of the discrete polymer when the random environment is heavy tailed.