  We study the directed polymer model in dimension ${1+1}$ when the environment
is heavy-tailed, with a decay exponent $\alpha\in(0,2)$. We give all possible
scaling limits of the model in the weak-coupling regime, i.e., when the inverse
temperature temperature $\beta=\beta_n$ vanishes as the size of the system $n$
goes to infinity. When $\alpha\in(1/2,2)$, we show that all possible
transversal fluctuations $\sqrt{n} \leq h_n \leq n$ can be achieved by tuning
properly $\beta_n$, allowing to interpolate between all super-diffusive scales.
Moreover, we determine the scaling limit of the model, answering a conjecture
by Dey and Zygouras [cf:DZ] - we actually identify five different regimes. On
the other hand, when $\alpha<1/2$, we show that there are only two regimes: the
transversal fluctuations are either $\sqrt{n}$ or $n$. As a key ingredient, we
use the Entropy-controlled Last Passage Percolation (E-LPP), introduced in a
companion paper [cf:BT_ELPP].
