  In recent years, experiments by several groups have demonstrated spontaneous
coherence in polariton systems, which can be viewed as a type of nonequilibrium
Bose-Einstein condensation. In these systems, the polariton lifetime is longer
than, but not much longer than, the polariton-polariton scattering time which
leads to the thermalization. By contrast, over the past twenty years several
groups have pursued experiments in a different system, which has very long
exciton lifetime, up to 30 microseconds or more, essentially infinite compared
to the thermalization time of the excitons. Thermal equilibrium of this type of
exciton in a trap has been demonstrated experimentally. In this system, the
interactions between the excitons are not short-range contact interactions, but
instead are dipole-dipole interactions, with the force at long range going as
$1/r^{3}$. Up to now there has not been a universally accepted demonstration of
BEC in this type of system, and the way forward will require better
understanding of the many-body effects of the excitons. I review what has been
learned and accomplished in the past two decades in the search for an
equilibrium BEC in this promising system.
