Quiescent and active galactic nuclei as factories of merging compact objects in the era of gravitational-wave astronomy

Galactic nuclei harbouring a central supermassive black hole (SMBH), possibly surrounded by a dense nuclear cluster (NC), represent extreme environments which house a complex interplay of many physical processes that uniquely affect stellar formation, evolution, and dynamics. The discovery of gravitational waves (GW) emitted by merging black holes (BHs) and neutron stars (NSs), funnelled a huge amount of work focused on understanding how compact object binaries (COBs) can pair-up and merge together. Here, we review from a theoretical standpoint how different mechanisms concur to the formation, evolution, and merger of COBs around quiescent SMBHs and active galactic nuclei (AGNs), summarizing the main predictions for current and future (GW) detections and outlining the possible features that can clearly mark a galactic nuclei origin.