  We review the available estimates of the masses of the compact object in
Ultraluminous X-ray Sources (ULXs) and critically reconsider the stellar-mass
versus intermediate-mass black hole interpretations. Black holes of several
hundreds to thousands of $M_\odot$ are not required for the majority of ULXs,
although they might be present in the handful of known hyper-luminous ($\sim
10^{41}$ erg s$^{-1}$) objects and/or some sources showing timing features in
their power density spectra. At the same time, however, stellar mass BHs may be
quite a reasonable explanation for ULXs below $\sim 10^{40}$ erg s$^{-1}$, but
they need super-Eddington accretion and some suitable dependence of the beaming
factor on the accretion rate in order to account for ULXs above this
(isotropic) luminosity. We investigate in detail a 'third way' in which a
proportion of ULXs contain $\approx 30-90 M_\odot$ black holes formed in a low
metallicity environment and accreting in a slightly critical regime and find
that it can consistently account for the properties of bright ULXs. Surveys of
ULX locations looking for a statistically meaningful relationship between ULX
position, average luminosity and local metallicity will provide a definitive
test of our proposal.
