Quantum spin chains provide some of the mathematically most accessible examples of quantum many-body systems. However, even these toy models pose considerable analytical and numerical challenges, due to the fact that the number of degrees of freedom involved grows exponentially fast with the system’s size. We will discuss the recent progress in establishing many body localization at the bottom of the spectrum of a disordered XXZ chain. In particular, we will introduce a new approach to many body localization that works beyond the droplet phase (joint work in progress with A. Klein).