  The observation of thermal emission from isolated neutron stars and the
modeling of the corresponding cooling curves has been very useful to get
information on the properties of matter at very high densities. More recently,
the detection of quiescent thermal emission from neutron stars in low mass
X-ray binary systems after active periods opened a new window to the physics of
matter at lower densities. Here we analyze a few sources that have been
recently monitored and we show how the models can be used to establish
constraints on the crust composition and their transport properties, depending
on the astrophysical scenarios assumed.
