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While neutron stars were first posited in the early 1930's, and discovered as pulsars in late 1960's it is only recently that we are beginning to understand the matter they contain. The importance of understanding their behavior and structure has been underlined by the recent observational discovery of gravitational radiation, since merging of neutron stars with black holes or neutron stars will be principal sources of future observations of gravitational radiation. In this talk I will describe the ongoing development of a consistent picture of the liquid interiors of neutron stars, driven by three recent advances: observations of heavy neutron stars with masses ~2.0 solar masses; determinations of masses and radii simultaneously for an increasing number of neutron stars; and an emerging understanding in QCD of how nuclear matter can turn into deconfined quark matter in the interior.