A detailed, rigorous study of the statistical mechanics-screening-and critical properties, phase diagrams, etc., of classical Coulomb monopole and dipole gases in two or more dimensions is presented. The statistical mechanics of the two-dimensional XY and Villain models is reconsidered and related to the one of two-dimensional lattice Coulomb gases. At low temperatures and moderate densities those gases behave like dipole gases. The Kosterlitz-Thouless transition is analyzed in that perspective and characterized by an order parameter. Techniques useful for a proof of existence of such a transition in a twodimensional hard-core Coulomb gas are developed and applied to the study of dipole gases.