The Erdös-Renyi graph G(N,d/N) is well known to undergo a dramatic change of behaviour around d of order log N. Below this threshold the graph develops inhomogeneities, which result in the emergence of a localized phase, in coexistence with a delocalized phase. I give an overview of the corresponding phase diagram, along with recent progress in establishing it rigorously. Joint work with Johannes Alt and Raphael Ducatez.