  Metasurfaces are subwavelength structured thin films consisting of arrays of
units that allow the controls of polarization, phase and amplitude of light
over a subwavelength thickness. The recent developments in topological
photonics have greatly broadened the horizon in designing the metasurfaces for
novel functional applications. In this review, we summarize recent progress in
the research field of topological metasurfaces, firstly from the perspectives
of passive and active in the classical regime, and then in the quantum regime.
More specifically, we begin by examining the passive topological phenomena in
two-dimensional photonic systems, including both time-reversal broken systems
and time-reversal preserved systems. Subsequently, we move to discuss the
cutting-edge studies of the active topological metasurfaces, including
nonlinear topological metasurfaces and reconfigurable topological metasurfaces.
After overviewing the topological metasurfaces in the classical regime, we show
how the topological metasurfaces could provide a new platform for quantum
information and quantum many-body physics. Finally, we conclude and describe
some challenges and future directions of this fast-evolving field.
