  Matter in neutron star cores reaches extremely high densities, forming states
of matter that cannot be generated in the laboratory. The Equation of State
(EOS) of the matter links to macroscopic observables, such as mass M and radius
R, via the stellar structure equations. A promising technique for measuring M
and R exploits hotspots (burst oscillations) that form on the stellar surface
when material accreted from a companion star undergoes a thermonuclear
explosion. As the star rotates, the hotspot gives rise to a pulsation, and
relativistic effects encode information about M and R into the pulse profile.
However the burst oscillation mechanism remains unknown, introducing
uncertainty when inferring the EOS. I review the progress that we are making
towards cracking this long-standing problem, and establishing burst
oscillations as a robust tool for measuring M and R. This is a major goal for
future large area X-ray telescopes.
