Ergodicity of many-particle motion is a fundamental assumption that underlies, among other things, the powerful statistical mechanics description of nature. The ergodic hypothesis can break down under some conditions, most notably in the presence of strong random potentials leading the phenomenon of Anderson localization. The theory of Anderson localization assumes no interactions among particles and it is of considerable practical interest to know whether the phenomenon can persist more generally. I review some recent ideas and results on spectral and transport properties of quantum and classical many-body systems. Within limitations of our methods we observe that localized states of classical particles are unstable against non-linearities, while interacting quantum particles can remain insulating.