We have conducted Chandra observations of $2560 arcmin 2 ($131 kpc 2 ) of M31 and find that the most luminous X-ray sources in most of our fields are in globular clusters. Of the 28 globular cluster X-ray sources in our fields, 15 are newly discovered. Approximately one-third of all the sources have L X ([0.5-7] keV Þ > 10 37 ergs s À1 , and approximately one-tenth of all the sources have L X ([0.5-7] keV) close to or above 10 38 ergs s À1 . The most luminous source, in the globular cluster Bo 375, is consistently observed to have L X greater than 2 Â 10 38 ergs s À1 . (1) We present data on the spectra and/or light curves of the five most luminous M31 globular cluster 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 timescale of the donor star. Our model can produce luminosities of several times 10 38 ergs s À1 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 À1 constitute a larger fraction of all GC sources than in our Galaxy. (4) We study the possible reasons for this difference between M31 and Galactic globular cluster X-ray sources and identify three promising explanations.