In this talk we will discuss host galaxy observations of Fast X-ray Transients (FXTs). FXTs are minute-to-hours long flashes of X-rays, discovered serendipitously in X-ray satellite data (mainly Chandra and XMM-Newton) and they have been associated with energetic extra-galactic phenomena. However, their exact nature has not yet been established. Although only about 30 are known, these extra-galactic FXTs are ubiquitous: at the sensitivity of Chandra and XMM-Newton there are about 10^6 FXTs all-sky per year. These FXTs are proven not to be caused by M-star flares. FXTs have been proposed to arise from double neutron star mergers, tidal disruption events involving an intermediate-mass black hole and a white dwarf, and from off-axis or sub-luminous gamma-ray bursts. Brief extra-galactic FXTs also arise in supernova shock breakouts. Contemporaneous multi-wavelength detections do not exist, this can be explained by the long time between the onset of the FXT and their discovery in the X-ray data. However, host galaxy redshifts have shown that they have luminosities ranging from 10^45-47 erg/s. In addition, the offset to the host galaxy is a powerful probe of their origin. The current host galaxy studies with 8- and 10-meter telescopes, for instance HiPERCAM mounted on the GTC, will in the near future be supplemented by Euclid observations.