Neutron stars contain the densest form of matter in the present universe. General relativity and causality set important constraints to their compactness. In addition, analytic GR solutions are useful in understanding the relationships that exist among the maximum mass, radii, moments of inertia, and tidal Love numbers of neutron stars, all of which are accessible to observation. Some of these relations are independent of the underlying dense matter equation of state, while others are very sensitive to the equation of state. Recent observations of neutron stars from pulsar timing, quiescent X-ray emission from binaries, and Type I X-ray bursts can set important constraints on the structure of neutron stars and the underlying equation of state. In addition, measurements of thermal radiation from neutron stars has uncovered the possible existence of neutron and proton superfluidity/superconductivity in the core of a neutron star, as well as offering powerful evidence that typical neutron stars have significant crusts. These observations impose constraints on the existence of strange quark matter stars, and limit the possibility that abundant deconfined quark matter or hyperons exist in the cores of neutron stars.