We have conducted Chandra observations of \sim 2560 square arcmin ( \sim 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 1 / 3 of all the sources have L _ { X } ( [ 0.5–7 ] keV ) ~ { } > 10 ^ { 37 } ergs s ^ { -1 } ; approximately 1 / 10 of all the sources have L _ { X } ( [ 0.5–7 ] keV ) close to or above 10 ^ { 38 } ergs s ^ { -1 } .
The most luminous source , in the globular cluster Bo 375 , is consistently observed to have L _ { X } greater than 2 \times 10 ^ { 38 } ergs s ^ { -1 } .

( 1 ) We present data on the spectra and/or light curves of the 5 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 time scale of the donor star .
Our model can produce luminosities of several times 10 ^ { 38 } ergs s ^ { -1 } , 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 } erg s ^ { -1 } 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 .