  In this contribution we review the theory of integrability of quantum systems
in one spatial dimension. We introduce the basic concepts such as the
Yang-Baxter equation, commuting currents, and the algebraic Bethe ansatz. Quite
extensively we present the treatment of integrable quantum systems at finite
temperature on the basis of a lattice path integral formulation and a suitable
transfer matrix approach (quantum transfer matrix). The general method is
carried out for the seminal model of the spin-1/2 $XXZ$ chain for which
thermodynamic properties like specific heat, magnetic susceptibility and the
finite temperature Drude weight of the thermal conductivity are derived.
