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Galactic nuclei harbouring a central supermassive black hole (SMBH), possibly surrounded by a dense nuclear cluster (NC), represent extreme environments that house a complex interplay of many physical processes that uniquely affect stellar formation, evolution, and dynamics. The discovery of gravitational waves (GWs) 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 with the formation, evolution, and merger of COBs around quiescent SMBHs and active galactic nuclei (AGNs), summarising the main predictions for current and future (GW) detections and outlining the possible features that can clearly mark a galactic nuclei origin.