Abstract:Low mass X-ray binaries that show thermonuclear X-ray bursts are ideal targets for constraining the equation of state of neutron star matter. We make use of time resolved, high count rate X-ray spectra to measure the Eddington limits and the apparent radii of neutron stars, which lead to the measurement of their masses and radii with an independent distance information. I will summarize a systematic analysis of the entire archive of RXTE X-ray burst observations for a number of sources, which allows for a better perception of the systematic uncertainties in these measurements. I will discuss the results of the application of this method to several X-ray binaries including EXO 1745-248, 4U 1820-30, 4U 1608-52, KS 1731-260, and SAX J1748.9-2021. The individual measurements allowed us to put strong constrains on the equation of state of matter at supranuclear densities.