  We demonstrate the existence of a two-dimensional anomalous Floquet insulator
(AFI) phase: an interacting (periodically-driven) non-equilibrium topological
phase of matter with no counterpart in equilibrium. The AFI is characterized by
a many-body localized bulk, exhibiting nontrivial micromotion within a driving
period, and delocalized (thermalizing) chiral states at its boundaries. For a
geometry without edges, we argue analytically that the bulk may be many-body
localized in the presence of interactions, deriving conditions where stability
is expected. We investigate the interplay between the thermalizing edge and the
localized bulk via numerical simulations of an AFI in a geometry with edges. We
find that non-uniform particle density profiles remain stable in the bulk up to
the longest timescales that we can access, while the propagating edge states
persist and thermalize, despite being coupled to the bulk. These findings open
the possibility of observing quantized edge transport in interacting systems at
high temperature. The analytical approach introduced in this paper can be used
to study the stability of other anomalous Floquet phases.
