  The remarkable discovery of kilohertz quasi-periodic brightness oscillations
(QPOs) in the accretion-powered emission from some sixteen neutron-star
low-mass X-ray binary systems has led to much speculation about and theoretical
modeling of the origin of these oscillations. It has also led to intense study
of the implications that these QPOs have for the properties of neutron stars
and of the accretion flow onto them. In this review we describe the strengths
and weaknesses of the models that have been proposed for the kilohertz QPOs
observed in the accretion-powered emission. We conclude that beat-frequency
models, and in particular the sonic-point model, are currently the most
promising. If these models are correct, the kilohertz QPOs provide the
strongest constraints to date on the masses and radii of neutron stars in
low-mass X-ray binaries, and on the equation of state of the dense matter in
all neutron stars.
