In this review, we consider a quantum Coulomb fluid made of charged point particles (typically electrons and nuclei). We describe various formalisms which start from the first principles of statistical mechanics. These methods allow systematic calculations of the equilibrium quantities in some particular limits. The effective-potential method is evocated first, as well as its application to the derivation of low-density expansions. We also sketch the basic outlines of the standard many-body perturbation theory. This approach is well suited for calculating expansions at high density (for Fermions) or at high temperature. Eventually, we present the Feynman-Kac path integral representation which leads to the introduction of an auxiliary classical system made of extended objects, i.e., filaments (also called “polymers”). The familiar Abe-Meeron diagrammatic series are then generalized in the framework of this representation. The truncations of the corresponding virial-like expansions provide equations of state which are asymptotically exact in the low-density limit at fixed temperature. The usefulness of such equations for describing the inner regions of the sun is briefly illustrated.