  Observation of cooling neutron stars can potentially provide information
about the states of matter at supernuclear densities. We review physical
properties important for cooling such as neutrino emission processes and
superfluidity in the stellar interior, surface envelopes of light elements due
to accretion of matter and strong surface magnetic fields. The neutrino
processes include the modified Urca process, and the direct Urca process for
nucleons and exotic states of matter such as a pion condensate, kaon
condensate, or quark matter. The dependence of theoretical cooling curves on
physical input and observations of thermal radiation from isolated neutron
stars are described. The comparison of observation and theory leads to a
unified interpretation in terms of three characteristic types of neutron stars:
high-mass stars which cool primarily by some version of the direct Urca
process; low-mass stars, which cool via slower processes; and medium-mass
stars, which have an intermediate behavior. The related problem of thermal
states of transiently accreting neutron stars with deep crustal burning of
accreted matter is discussed in connection with observations of soft X-ray
transients.
