All electrons are identical and subject to the same forces, yet the materials they make up possess a dazzling diversity of physical properties. The interplay of electron-electron interaction and random disorder is largely responsible for this emergent complexity. This talk presents two examples of non-trivial behavior arising from simple theoretical models of electronic systems in low dimensions. The first part of the talk is on the “Dyson’’ model, in which non-interacting electrons on a disordered 1-D chain hop randomly between nearest neighbors in the absence of an on-site potential. It is shown that a choice of singular hopping distributions induces continuously tunable non-universal behavior in the localization length and density of states at the band center. The second part of this talk is about the effect of disorder and a carefully tuned periodic potential on the quantum Hall effect of a 2-D interacting electron gas in a high magnetic field. Here, the single particle spectrum resolves into nearly flat Landau level subbands of varying topological character. Topologically trivial subbands, with Chern number zero, may host a many-body localized (MBL) phase at large disorder. However, MBL is absent topologically nontrivial subbands, with nonzero Chern number. This approach enables one to disentangle the roles of topology and dimensionality in destabilizing MBL in the quantum Hall system. These examples illustrate how a combination of theory and computation can reveal new insights even in relatively well-studied topics in condensed matter physics.