Abstract: It has been over fifty years since Anderson's original paper reporting the discovery of the absence of diffusion in random lattices, and over three decades since the metal-insulator transition was understood in terms of the scaling theory of localization. Despite this long time interval, and thousands of publications on the subject, the Anderson model of localization continues to offer new insights. In this talk we examine the Anderson model at large disorder (i.e. in the insulating phase). We find, surprisingly, new singularities in this regime, which had apparently escaped attention. These singularities appear to demarcate the boundary between "typical" and "rare fluctuation" states. (The latter are the counterpart of rare fluctuation effects found to be quite pervasive in quantum many-body models with large disorder). Besides exposing a new facet of a seemingly simple and extensively studied model, our work suggests that Anderson's model of localization offers an unparalleled opportunity to understand rare fluctuation effects at a level of detail that has not been possible in numerical approaches to many-body models.