In a disordered potential, the diffusive transport of non-interacting particles can be inhib- ited, a phenomenon known as Anderson localization. In three dimensions, there exists a quantum phase-transition between localized (insulator) and diffusive (metal) dynamics. A long-standing question is the effect of interactions on such dynamics. The goal of this thesis is to investigate this problem theoretically and numerically in the experimental framework of Bose-Einstein condensates. In one dimension, the interplay between disorder and in- teractions leads to the existence of three regimes which are characterized with a spectral approach. In three dimensions, using a "quantum simulator" of the 3D Anderson model, we show the emergence of sub-diffusion in lieu of Anderson localization. Considering the excitations of the system in the very weakly interacting regime, we also demonstrate that the concept of universality of the Anderson transition also applies to Bogoliubov quasi-particles. Finally, we show the relevance of a new method, the Truncated Husimi method, in order to take into account the effect of quantum noise on interacting disordered systems.
