  I review the basic phenomenology and theory of the millisecond brightness
oscillations observed during thermonuclear X-ray bursts from 13 of
approximately 70 accreting neutron stars in low-mass X-ray binaries. Compelling
observations indicate that the oscillations are produced by surface brightness
patterns on the rapidly rotating neutron stars. However, it remains to be
understood (1) why the brightness patterns producing them persist for up to 15
s during an X-ray burst, whereas the burning should cover the entire surface in
less than 1 s, and (2) why the frequencies drift upward by about 5 Hz during
the course of the burst. These peculiarities can probably be explained by
taking into account the expansion of the surface layers caused by the burning,
zonal flows that form due to pressure gradients between the equator and poles,
and Rossby-Alfven modes that are excited in the surface ocean. Further progress
toward understanding how burning progresses on the surface of the neutron star
can be made with a next-generation X-ray timing mission, which would provide a
larger sample of sources with oscillations, detect sideband signals produced by
the spectrum of modes that should be excited in the neutron star ocean, and
measure harmonic structure in the profiles of the oscillations. These
observations would be crucial for measuring the distribution of the rotation
rates of neutron stars, the progression of unstable nuclear burning in the
accreted ocean, and the curvature of the space-time around the neutron star.
