  Recently, several ultraluminous X-ray (ULX) sources were shown to host a
neutron star (NS) accretor. We perform a suite of evolutionary calculations
which show that, in fact, NSs are the dominant type of ULX accretor. Although
black holes (BH) dominate early epochs after the star-formation burst, NSs
outweigh them after a few 100 Myr and may appear as late as a few Gyr after the
end of the star formation episode. If star formation is a prolonged and
continuous event (i.e., not a relatively short burst), NS accretors dominate
ULX population at any time in solar metallicity environment, whereas BH
accretors dominate when the metallicity is sub-solar. Our results show a very
clear (and testable) relation between the companion/donor evolutionary stage
and the age of the system. A typical NS ULX consists of a $\sim1.3\,M_\odot$ NS
and $\sim1.0\,M_\odot$ Red Giant. A typical BH ULX consist of a
$\sim8\,M_\odot$ BH and $\sim6\,M_\odot$ main-sequence star. Additionally, we
find that the very luminous ULXs ($L_X\gtrsim10^{41}$ erg/s) are predominantly
BH systems ($\sim9\,M_\odot$) with Hertzsprung gap donors ($\sim2\,M_\odot$).
Nevertheless, some NS ULX systems may also reach extremely high X-ray
luminosities ($\gtrsim10^{41}$ erg/s).
