When studying disordered systems, the influence of disorder on the phase transition is a central question: one wants determine whether an arbitrary quantity of disorder modifies the critical properties of the system, with respect to the non-disordered case. We present here an overview of the mathematical results obtained to answer that question for the polymer pinning model. In the IID case, the picture of disorder relevance/irrelevance is by now established, and follows the so-called Harris

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... terion: disorder is irrelevant if ν hom > 2 and relevant if ν hom < 2, where ν hom is the order of the homogeneous phase transition. In the correlated case, Weinrib and Halperin predicted that, if the two point correlation decays as a power law with exponent a > 0, then the Harris criterion would be modified if a < 1: disorder should be relevant whenever ν hom < 2 max(1, 1/a). It turns out that this prediction is not accurate: the key quantity is not the decay exponent a, but the occurrence of rare regions with atypical disorder. An infinite disorder regime may appear, in which the relevance / irrelevance picture is crucially modified. We also mention another recent approach to the question of the influence of disorder for the pinning model: the persistence of disorder when taking the scaling limit of the system.