  We investigate the thermal stability of nuclear burning on the accreting
X-ray pulsar GRO J1744-28. The neutron star's dipolar magnetic field is
<3\times 10^{11} G if persistent spin-up implies that the magnetospheric radius
is less than the co-rotation radius. After inferring the properties of the
neutron star, we study the thermal stability of hydrogen/helium burning and
show that thermonuclear instabilities are unlikely causes of the hourly bursts
seen at very high accretion rates. We then discuss how the stability of the
thermonuclear burning depends on both the global accretion rate and the neutron
star's magnetic field strength. We emphasize that the appearance of the
instability (i.e., whether it looks like a Type I X-ray burst or a flare
lasting a few minutes) will yield crucial information on the neutron star's
surface magnetic field and the role of magnetic fields in convection. We
suggest that a thermal instability in the accretion disk is the origin of the
long (~300 days) outburst and that the recurrence time of these outbursts is
>50 years. We also discuss the nature of the binary and point out that a
velocity measurement of the stellar companion (most likely a Roche-lobe filling
giant with m_K>17) will constrain the neutron star mass.
