Luminous supersoft X-ray sources were discovered with the Einstein observatory and have been established as an important new class of X-ray binaries on the basis of observations with the Roentgen Satellite (ROSAT). They have extremely soft spectra (equivalent blackbody temperatures of ∼15-80 eV) and are highly luminous (bolometric luminosities of 10 36 -10 38 erg s −1 ). Correcting for the heavy extinction of soft X rays by interstellar neutral hydrogen, their numbers in the disks of ordinary spiral galaxies like our own and M31 are estimated to be of the order of 10 3 . Their observed characteristics are consistent with those of white dwarfs, which are steadily or cyclically burning hydrogen-rich matter accreted onto the surface at a rate of order 10 −7 M year −1 . The required high accretion rates can be supplied by mass transfer on a thermal time scale (10 6 -10 7 years) from close companion stars that are more massive than the white dwarf accretor, typically 1.3-2.5 M . Steady burning can also occur in a post-nova stage, but for shorter time scales, and it has been observed in a few classical novae and symbiotic novae. A few supersoft sources have been found to be recurrent transients. They are possibly connected with very massive white dwarfs accreting at high rates. Luminous supersoft sources may make a considerable contribution to the Type Ia supernova rate in spiral and irregular galaxies.