Many-body methods at finite temperature

The purpose of the present paper is to review the approximation methods relevant to describe many-fermion systems at finite temperature. In Sections 1.2 and 1.3 we review the grand canonical formalism for independent fermions and discuss its applicability to the case of finite nuclei for which fluctuations arising from the small number of particles involved are expected to be sizeable. In Section 1.4 we present a derivation of the mean-field equations based on the variational method. In Section 2 we discuss perturbation expansions of partition functions. We consider a particularly important subseries containing the so called ring diagrams whose summation leads to the random phase approximation (RPA). In Section 3 an application to the physics of giant resonances in hot nuclei is described