Observations of hundreds of supersoft x-ray sources (SSSs) in external galaxies have shed light on the diversity of the class and on the natures of the sources. SSSs are linked to the physics of Type Ia supernovae and accretion-induced collapse, ultraluminous x-ray sources and black holes, the ionization of the interstellar medium, and tidal disruption by supermassive black holes. The class of SSSs has an extension to higher luminosities: ultraluminous SSSs have luminosities above 10^39 erg/s. There is also an extension to higher energies: quasisoft x-ray sources (QSSs) emit photons with energies above 1 eV, but few or none with energies above 2 keV. Finally, a significant fraction of the SSSs found in external galaxies switch states between observations, becoming either quasisoft or hard. For many systems ``supersoft refers to a temporary state; SSSs are sources, possibly including a variety of fundamentally different system types, that pass through such a state. We review those results derived from extragalactic data and related theoretical work that are most surprising and that suggest directions for future research.
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