Neutron stars are among the densest objects in the universe. The conditions in their centers are largely unconstrained by current theoretical physics or terrestrial laboratories, leaving a wide variety of compositions and structures possible. Observations of thermal emission from neutron stars -- specifically measurements of their sizes and cooling rates -- may therefore be the best way to constrain the behavior of matter in these extreme conditions. I will discuss a sample of nearby, cooling neutron stars that we are using for this purpose. We are attempting to pin down the basic parameters of these neutron stars with a variety of ground- and space-based observations, coupled with theoretical modeling. Along the way, we have encountered a number of interesting astrophysical puzzles that I will describe.