I will present Chandra and XMM-Newton observations of several neutron star X-ray transients during their quiescent states. In particular, I will discuss very recent results obtained on two systems: KS 1731-260 and MXB 1659-29. In contrast to normal transients which accrete only for a few weeks to a few months when they are in outbursts, KS 1731-260 was accreting for over 12 years before returning to quiescence and MXB 1659-29 for 2.5 years. These prolonged episodes of actively accreting had dramatic effects on the state of the crust of the neutron stars in those systems. My recent Chandra and XMM-Newton observations have allowed me to follow the cooling of the neutron star crusts in those systems which show that the crustal heat conductivity should be high. Furthermore, the temperature of the core of the neutron stars was found to be very cold, likely due to enhanced cooling processes in the cores. I will discuss these findings in the context of our understanding of the super-dense matter in neutron stars.