Abstract:Anderson localization is an intriguing illustration of a disorder-driven phenomenon in which, in the presence of sufficient disorder, a non-interacting quantum system can become diffusion-less. For almost 50 years people believed that interactions would delocalize an Anderson localized system and only around 15 years back a perturbative calculation revealed that localization can persist even in the presence of interactions, a phenomenon known as Many-body localization (MBL) which is an interesting and unusual phenomenon in many aspects. An isolated quantum system in the MBL phase is non-ergodic and hence, challenges the basic foundations of quantum statistical physics. Local observables in the MBL phase do not thermalize leading to the violation of eigenstate thermalization hypothesis. This results in a rich behaviour of entanglement entropy and a long time memory of the initial state in local observables. MBL has been proved to exist in one-dimensional systems and has also been observed in cold atom experiments. But the nature of the MBL transition is still an open question. In this talk, after providing a brief overview of the fundamental characterizations of the MBL phase, I will discuss some of our recent works on novel characterizations of the MBL phase, the nature of the MBL transition, stability of the MBL phase and potential for an intermediate non-ergodic extended phase in these disordered interacting quantum systems.