I will introduce the directed polymer model in dimension 1+d, which describes a one-dimensional polymer chain placed in some disordered medium. I will give an overview of the model and I will discuss its relation to the stochastic heat equation (SHE) with multiplicative noise. This has been widely studied in the case where the disorder admits a second moment ; it corresponds to the SHE with Gaussian white noise. I will then present the results obtained in collaboration with Carsten Chong (Columbia, USA) and Hubert Lacoin (IMPA, Brazil) on directed polymers and the SHE with Lévy noise.