Neutron stars are the most compact objects known to exist in the universe. Their masses have been measured to exquisite precision, but precise radius measurements have eluded us until recently. I will describe developments in the theory of neutron star emission in transient low mass X-ray binaries in quiescence, and recent observations of these systems with the X-ray observatories Chandra and XMM. Together, these lead to an observational path forward to precision neutron star radius measurements, useful for constraining the dense matter equation of state.