  We discuss the application of techniques of quantum estimation theory and
quantum metrology to thermometry. The ultimate limit to the precision at which
the temperature of a system at thermal equilibrium can be determined is related
to the heat capacity when global measurements are performed on the system. We
prove that if technical or practical limitations restrict our capabilities to
local probing, the highest achievable accuracy to temperature estimation
reduces to a sort of mesoscopic version of the heat capacity. Adopting a more
practical perspective, we also discuss the relevance of qubit systems as
optimal quantum thermometers, in order to retrieve the temperature, or to
discriminate between two temperatures, characterizing a thermal reservoir. We
show that quantum coherence and entanglement in a probe system can facilitate
faster, or more accurate measurements of temperature. While not surprising
given this has been demonstrated in phase estimation, temperature is not a
conventional quantum observable, so that these results extend the theory of
parameter estimation to measurement of non-Hamiltonian quantities. Finally we
point out the advantages brought by a less standard estimation technique based
on sequential measurements, when applied to quantum thermometry.
