  We discuss pycnonuclear burning of highly exotic atomic nuclei in deep crusts
of neutron stars, at densities up to 1e13 g/cc. As an application, we consider
pycnonuclear burning of matter accreted on a neutron star in a soft X-ray
transient (SXT, a compact binary containing a neutron star and a low-mass
companion). The energy released in this burning, while the matter sinks into
the stellar crust under the weight of newly accreted material, is sufficient to
warm up the star and initiate neutrino emission in its core. The surface
thermal radiation of the star in quiescent states becomes dependent of poorly
known equation of state (EOS) of supranuclear matter in the stellar core, which
gives a method to explore this EOS. Four qualitatively different model EOSs are
tested against observations of SXTs. They imply different levels of the
enhancement of neutrino emission in massive neutron stars by (1) the direct
Urca process in nucleon/hyperon matter; (2) pion condensates; (3) kaon
condensates; (4) Cooper pairing of neutrons in nucleon matter with the
forbidden direct Urca process. A low level of the thermal quiescent emission of
two SXTs, SAX J1808.4-3658 and Cen X-4, contradicts model (4). Observations of
SXTs test the same physics of dense matter as observations of thermal radiation
from cooling isolated neutron stars, but the data on SXTs are currently more
conclusive.
