The article reviews the two-dimensional Coulomb gas model and its connection to vortex fluctuations for a two-dimensional superfluid. The neutral and non-neutral versions of the Coulomb gas are discussed and the relation to an equivalent sine-Gordon field theory is given. The charge-unbinding picture is used to elucidate some essential properties of the Coulomb gas. Derivations of approximate renormalization equations are sketched and the phase transition for the neutral two-dimensional Coulomb gas is described. The Kosterlitz renormalization-group equations are reviewed in some detail. The vortex-Coulomb gas charge analogy is carefully explained. The connections between experiments for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ films and superconducting films and the neutral and non-neutral versions of the Coulomb gas are outlined using concepts like the universal jump and the Coulomb gas scaling relations. The properties of a dynamical version of the Coulomb gas, corresponding to vortex dynamics, are discussed and related to experiments. An analogy with Maxwell's equations in two dimensions is also given.

DOI:https://doi.org/10.1103/RevModPhys.59.1001