It has been argued that recent data from the Chandra and XMM-Newton X-ray telescopes on cooling neutron stars suggest the presence of an exotic phase of matter in these objects. Given the importance of this interpretation, we consider in detail the cooling of neutron stars within a "Minimal Cooling" picture, i.e. neutron stars containing only neutrons and a small fraction of protons (+ leptons) in their cores. This scenario, which is similar to but replaces the so-called "Standard Cooling" paradigm, is augmented by the plausible "standard" physics of nucleon pairing. It is the benchmark against which data have to be compared before any claim of evidence for some more exotic phase or cooling process can be made. Within the Minimal Cooling picture, the direct Urca process cannot operate because either the proton fraction remains small or exotic matter in the form of hyperons, Bose condensates, and/or quark matter does not appear. This minimal picture is well-defined in terms of the equation of state and the neutron star mass and radius, but contains major uncertainties related with the theoretical predictions of nucleon pairing in the core and the chemical composition of the upper layers of the star (the envelope, but not the atmosphere). The dominant neutrino emission mechanism which drives the cooling of young (less than 10^5 years old) stars is the formation and destruction of Cooper pairs, which is sensitive to nucleon pairing, and not the Modified Urca process. [Ongoing work in collaboration with J.M. Lattimer, M. Prakash and A. Steiner].