  A new classification of neutron star cooling scenarios, involving either
``minimal'' cooling or ``enhanced'' cooling is proposed. The minimal cooling
scenario replaces and extends the so-called standard cooling scenario to
include neutrino emission from the Cooper pair breaking and formation process.
This emission dominates that due to the modified Urca process for temperatures
close to the critical temperature for superfluid pairing. Minimal cooling is
distinguished from enhanced cooling by the absence of neutrino emission from
any direct Urca process, due either to nucleons or to exotica. Within the
minimal cooling scenario, theoretical cooling models can be considered to be a
four parameter family involving the equation of state of dense matter,
superfluid properties of dense matter, the composition of the neutron star
envelope, and the mass of the neutron star. Consequences of minimal cooling are
explored through extensive variations of these parameters. Results are compared
with the inferred properties of thermally-emitting neutron stars in order to
ascertain if enhanced cooling occurs in any of them. All stars for which
thermal emissions have been clearly detected are at least marginally consistent
with the lack of enhanced cooling. The two pulsars PSR 0833-45 (Vela) and PSR
1706-44 would require enhanced cooling in case their ages and/or temperatures
are on the lower side of their estimated values whereas the four stars PSR
0656+14, PSR 1055-52, Geminga, and RX J0720.4-3125 may require some source of
internal heating in case their age and/or luminosity are on the upper side of
their estimated values. The new upper limits on the thermal luminosity of PSR
J0205+6449 and RX J0007.0+7302 are indicative of the occurrence of some
enhanced neutrino emission beyond the minimal scenario.
