  Ultraluminous X-ray sources (ULXs) with Lx > 10^{39} ergs/s have been
discovered in great numbers in external galaxies with ROSAT, Chandra, and XMM.
The central question regarding this important class of sources is whether they
represent an extension in the luminosity function of binary X-ray sources
containing neutron stars and stellar-mass black holes (BHs), or a new class of
objects, e.g., systems containing intermediate-mass black holes (100-1000
Msun). We have carried out a theoretical study to test whether a large fraction
of the ULXs, especially those in galaxies with recent star formation activity,
can be explained with binary systems containing stellar-mass black holes. To
this end, we have applied a unique set of binary evolution models for
black-hole X-ray binaries, coupled to a binary population synthesis code, to
model the ULXs observed in external galaxies. We find that for donor stars with
initial masses >10 Msun the mass transfer driven by the normal nuclear
evolution of the donor star is sufficient to potentially power most ULXs. This
is the case during core hydrogen burning and, to an even more pronounced
degree, while the donor star ascends the giant branch, though the latter phases
lasts only ~5% of the main sequence phase. We show that with only a modest
violation of the Eddington limit, e.g., a factor of ~10, both the numbers and
properties of the majority of the ULXs can be reproduced. One of our
conclusions is that if stellar-mass black-hole binaries account for a
significant fraction of ULXs in star-forming galaxies, then the rate of
formation of such systems is ~3 x 10^{-7} per year normalized to a
core-collapse supernova rate of 0.01 per year.
