  We review the nuclear and condensed matter physics underlying the thermal and
transport properties of the neutron star inner crust. These properties play a
key role in interpreting transient phenomena such as thermal relaxation in
accreting neutron stars, superbursts, and magnetar flares. We emphasize
simplifications that occur at low temperature where the inner crust can be
described in terms of electrons and collective excitations. The heat
conductivity and heat capacity of the solid and superfluid phase of matter is
discussed in detail and we emphasize its role in interpreting observations of
neutron stars in soft X-ray transients. We highlight recent theoretical and
observational results, and identify future work needed to better understand a
host of transient phenomena in neutron stars.
