  We discuss the problem of localization in two dimensional electron systems in
the quantum Hall (single Landau level) regime. After briefly summarizing the
well-studied problem of Anderson localization in the non-interacting case, we
concentrate on the problem of disorder induced many-body localization (MBL) in
the presence of electron-electron interactions using numerical exact
diagonalization and eigenvalue spacing statistics as a function of system size.
We provide evidence showing that MBL is not attainable in a single Landau level
with short range (white noise) disorder in the thermodynamic limit. We then
study the interplay of topology and localization, by contrasting the behavior
of topological and nontopological subbands arising from a single Landau level
in two models - (i) a pair of extremely flat Hofstadter bands with an optimally
chosen periodic potential, and (ii) a Landau level with a split-off
nontopological impurity band. Both models provide convincing evidence for the
strong effect of topology on the feasibility of many-body localization as well
as slow dynamics starting from a nonequilibrium state with charge imbalance.
