Photonic metasurfaces, that is, two-dimensional arrangements of designed plasmonic or dielectric resonant scatterers, have been established as a successful concept for controlling light fields at the nanoscale. While the majority of research so far has concentrated on passive metasurfaces, the direct integration of nanoscale emitters into the metasurface architecture offers unique opportunities ranging from fundamental investigations of complex light-matter interactions to the creation of flat sources of tailored light fields. While the integration of emitters in metasurfaces as well as many fundamental effects occurring in such structures were initially studied in the realm of nanoplasmonics, the field has recently gained significant momentum following the development of Mie-resonant dielectric metasurfaces. Because of their low absorption losses, additional possibilities for emitter integration, and compatibility with semiconductor-based light-emitting devices, all-dielectric systems are promising for highly efficient metasurface light sources. Furthermore, a flurry of new emission phenomena are expected based on their multipolar resonant response. This review reports on the state of the art of light-emitting metasurfaces, covering both plasmonic and all-dielectric systems.