We investigate models for the class of ultraluminous nonnuclear X-ray sources (i.e., ultraluminous compact X-ray sources [ULXs]) seen in a number of galaxies and probably associated with star-forming regions. Models in which the X-ray emission is assumed to be isotropic run into several difficulties. In particular, the formation of sufficient numbers of the required ultramassive black hole X-ray binaries is problematic, and the likely transient behavior of the resulting systems is not in good accord with observation. The assumption of mild X-ray beaming suggests instead that ULXs may represent a short-lived but extremely common stage in the evolution of a wide class of X-ray binaries. The best candidate for this is the phase of thermal-timescale mass transfer that is inevitable in many intermediate- and high-mass X-ray binaries. This in turn suggests a link with the Galactic microquasars. The short lifetimes of high-mass X-ray binaries would explain the association of ULXs with episodes of star formation. These considerations still allow the possibility that individual ULXs may contain extremely massive black holes.