Our overall goal was to better understand the effects of disorder on many-body quantum systems. While such effects have been frequently studied in physics (where they play an important role in a number of intriguing phenomena like Bose-Einstein condensation, superconductivity, and the quantum Hall effect), their mathematical understanding is much more limited. We attribute this to the multiple challenges one encounters in the study of interacting quantum many-body systems (ground states, thermal equilibrium states and phase transitions, dynamics) on the one hand, and random Schrödinger operators on the other hand. Almost inevitably, when one tries to understand the effect of disorder on many-body quantum systems, one is faced with a combination of diffculties already present in these two major areas of mathematical physics separately. In both areas, however, there has been significant progress in the past decade. For example, recent progress on quantum spin systems, which are simple models of many-body physics, is contained in works by Bravyi, Hastings, Matsui, Nachtergaele, Sims, Yarotsky and similarly, Aizenman, Bourgain, Germinet, Klein, Stolz, Warzel have contributed to modern advances in the theory of localization for random Schrödinger operators. As can be seen from the recent works by Chulaevsky and Suhov as well as Aizenman and Warzel, we now have reached the point where we can start to answer interesting questions about many-body systems with randomness.