Constraints on the equation of state of ultra-dense matter from observations of neutron stars

I discuss constraints on the equation of state of matter at supra-nuclear densities that can be derived from observations of neutron stars. I focus on recent work on Vela X-1, which may well be substantially more massive than the canonical 1.4 solar masses, and on the prospects offered by the isolated' or thermally-emitting’ neutron stars. I present a re-analysis of the LETG observations of the brightest of the isolated neutron stars, RX J1856.5-3754, and find that the spectrum shows, unlike all other neutron stars observed so far, a significant absorption feature, near 39 Angstrom.