### Abstract

Ever since the Wigner-Dyson ensembles of random matrices and the Anderson model of random Schroedinger operators, and more recently also quantum graphs, lessons from statistical mechanics have led to fundamental insights into the effects of disorder on spectra and dynamics of random operators, and indeed also on the effects of pseudo-randomness which reach as far afield as analytic number theory. The talk will mention some of the celebrated observations, outstanding conjectures, and recent developments in the mathematical analysis of localization and related gap statistics. Along with insights related to quantum dynamics, the field demonstrates again the long reach of some of the central themes of modern statistical mechanics - which need of course to be combined with mathematical tools which are specific to the subject.