Metasurfaces are subwavelength structured thin films consisting of arrays of units that allow the control of polarization, phase, and amplitude of light over a subwavelength thickness. Recent developments in topological photonics have greatly broadened the horizon in designing metasurfaces for novel functional applications. In this review, we summarize recent progress in the research field of topological metasurfaces, first 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 discuss the cutting-edge studies of active topological metasurfaces, including nonlinear topological metasurfaces and reconfigurable topological metasurfaces. After overviewing topological metasurfaces in the classical regime, we show how they 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.