The cores of neutron stars contain matter at densities beyond nuclear saturation. Understanding such matter therefore feeds into understanding of neutron stars, and vice versa; studies of neutron stars can yield unique information about the state of high-density matter. Here we summarize current constraints based on observations of neutron stars. We focus on observations of neutron star binaries, because mass measurements are easiest and most precise for such systems. We highlight recent developments in observation and modeling of neutron star low-mass X-ray binaries that suggest,that slowly rotating neutron stars may have maximum masses up to similar to 2.2 M-circle dot, implying a high-density equation of state that is relatively hard. [References: 33]
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