Neutron stars have long been regarded as extraterrestrial laboratories from which we can learn about extreme energy density matter at low temperatures. In this article, some of the recent advances made in astrophysical observations and related theory are highlighted. Although the focus is on the much needed information on masses and radii of several individual neutron stars, the need for additional knowledge about the many facets of neutron stars is stressed. The extent to which quark matter can be present in neutron stars is summarized with emphasis on the requirement of non-perturbative treatments. Some longstanding and new questions, answers to which will advance our current status of knowledge, are posed.