TeV Dark Matter Searches in the Extragalactic Gamma-Ray Sky

High-energetic gamma rays from astrophysical targets constitute a unique probe for annihilation or decay of heavy particle dark matter (DM). After several decades, diverse null detections have resulted in strong constraints for DM particle masses up to the TeV scale. While the gamma-ray signature is expected to be universal from various targets, uncertainties of astrophysical origin strongly affect and weaken the limits. At the same time, spurious signals may originate from non-DM related processes. The many gamma-ray targets in the extragalactic sky being searched for DM play a crucial role to keep these uncertainties under control and to ultimately achieve an unambiguous DM detection. Lately, a large progress has been made in combined analyses of TeV DM candidates towards different targets by using data from various instruments and over a wide range of gamma-ray energies. These approaches not only resulted in an optimal exploitation of existing data and an improved sensitivity, but also helped to level out target- and instrument-related uncertainties. This review gathers all searches in the extragalactic sky performed so far with the space-borne Fermi-Large Area Telescope, the ground-based imaging atmospheric Cherenkov telescopes, and the High-Altitude Water Cherenkov Gamma-Ray Observatory (HAWC). We discuss the different target classes and provide a complete list of all analyses so far.