  We study the statistical mechanics of a multicomponent two-dimensional
Coulomb gas which lives on a finite surface without boundaries. We formulate
the Debye--Huckel theory for such systems, which describes the low-coupling
regime. There are several problems, which we address, to properly formulate the
Debye--Huckel theory. These problems are related to the fact that the electric
potential of a single charge cannot be defined on a finite surface without
boundaries. One can only define properly the Coulomb potential created by a
globally neutral system of charges. As an application of our formulation, we
study, in the Debye--Huckel regime, the thermodynamics of a Coulomb gas living
on a sphere of radius $R$. We find, in this example, that the grand potential
(times the inverse temperature) has a universal finite-size correction $(1/3)
\ln R$. We show that this result is more general: for any arbitrary finite
geometry without boundaries, the grand potential has a finite-size correction
$(\chi/6) \ln R$, with $\chi$ the Euler characteristic of the surface and $R^2$
its area.
