With a mass similar to that of the Sun compressed inside a radius of about 10 km, neutron stars push matter to the extremes. At these high densities, the composition of matter (encoded in the so-called equation of state) depends on particle interactions in a regime that cannot be tested by collider experiments. In addition, the state of matter involves the full complement of concepts from low-temperature physics. The star’s outer layers have solidified into an elastic crust of neutron-rich nuclei, while the core contains an exotic superfluid-superconducting mixture. In this talk I will introduce the main issues involved in trying to understand these objects, paying particular attention to the relevance of superfluidity. I will outline how astrophysical observations, both electromagnetic and through gravitational waves, are helping us probe these exciting objects.