The main research theme of RANDOP revolves around the spectral study of certain random operators, most notably random Schrödinger operators. A central question in this field, originally motivated by physical considerations, is to determine the behavior of the (generalized) eigenfunctions of such operators: are they concentrated on small regions of space (localization phenomenon) or do they put (L^2)-mass everywhere (as for the Laplacian)? Building on our previous results on the topic, we aim at addressing ambitious questions on localization and delocalization phenomenon for several types of operators and at exploring their consequences for dynamical properties and random matrix theory.