We study the model of Directed Polymers in Random Environment in 1+1 dimensions, where the distribution at a site has a tail which decays regularly polynomially with power a, where a ? (0, 2). After proper scaling of temperature b-1, we show strong localization of the polymer to a favorable region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (a, b)-indexed family of measures on Lipschitz curves lying inside the 45-degrees-rotated square with unit diagonal. In particular, this shows order n transversal fluctuations of the polymer. If, and only if, a is small enough, we find that there exists a random critical temperature below which, but not above, the effect of the environment is macroscopic. The results carry over to d+1 dimensions for d > 1 with minor modifications. Joint work with Antonio Auffinger (NYU).