Burst oscillations, a phenomenon observed in a significant fraction of Type I (thermonuclear) X-ray bursts, involve the development of highly asymmetric brightness patches in the burning surface layers of accreting neutron stars. Intrinsically interesting as nuclear phenomena, they are also important as probes of dense matter physics and the strong gravity, high magnetic field environment of the neutron star surface. Burst oscillation frequency is also used to measure stellar spin, and doubles the sample of rapidly rotating (above 10 Hz) accreting neutron stars with known spins. Although the mechanism remains mysterious, burst oscillation models must take into account thermonuclear flame spread, nuclear processes, rapid rotation, and the dynamical role of the magnetic field. This review provides a comprehensive summary of the observational properties of burst oscillations, an assessment of the status of the theoretical models that are being developed to explain them, and an overview of how they can be used to constrain neutron star properties such as spin, mass, and radius.
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