  We have conducted Chandra observations of ~2560 square arcmin (~131 kpc^2) of
of M31, and find that the most luminous X-ray sources in our fields are in
globular clusters (GCs). Of the 28 GC X-ray sources in our fields, 17 are newly
discovered. Approximately 1/3 of all the sources have L_x([0.5-7] keV) >
10^{37} ergs/s; approximately 1/10 of all the sources have L_x([0.5-7] keV)
close to or above 10^{38} erg/s. The most luminous source, in the globular
cluster Bo 375, is consistently observed to have L_x greater than 2 x 10^{38}
ergs/s.
  (1) We present data on the spectra and/or light curves of the 5 most luminous
M31 GC sources.
  (2)We explore possible explanations for the high X-ray luminosities of the
brightest sources. These include that the X-ray sources may be composites, the
radiation we receive may be beamed, metallicity effects could be at work, or
the sources may be accreting black holes. We weigh each of these possibilities
against the data. In addition, we introduce a neutron star model in which mass
transfer proceeds on the thermal time scale of the donor star. Our model can
produce luminosities of several times 10^{38} ergs/s, and leads to a set of
well-defined predictions.
  (3) We compute the X-ray luminosity function and the distribution of counts
in wavebands that span the range of energies to which Chandra is sensitive. We
find the peak X-ray luminosity is higher and that systems with L_x > 10^{37}
ergs/s constitute a larger fraction of all GC sources than in the MW.
  (4) We study the possible reasons for this difference between M31 and
Galactic GC X-ray sources and identify three promising explanations.
