  We present two methods for solving the electrostatics of point charges and
multipoles on the surface of a sphere, \textit{i.e.} in the space
$\mathcal{S}_{2}$, with applications to numerical simulations of
two-dimensional polar fluids.
  In the first approach, point charges are associated with uniform neutralizing
backgrounds to form neutral pseudo-charges, while, in the second, one instead
considers bi-charges, \textit{i.e.} dumbells of antipodal point charges of
opposite signs. We establish the expressions of the electric potentials of
pseudo- and bi-charges as isotropic solutions of the Laplace-Beltrami equation
in $\mathcal{S}_{2}$. A multipolar expansion of pseudo- and bi-charge
potentials leads to the electric potentials of mono- and bi-multipoles
respectively. These potentials constitute non-isotropic solutions of the
Laplace-Beltrami equation the general solution of which in spherical
coordinates is recast under a new appealing form.
  We then focus on the case of mono- and bi-dipoles and build the theory of
dielectric media in $\mathcal{S}_{2}$. We notably obtain the expression of the
static dielectric constant of a uniform isotropic polar fluid living in
$\mathcal{S}_{2}$ in term of the polarization fluctuations of subdomains of
$\mathcal{S}_{2}$. We also derive the long range behavior of the equilibrium
pair correlation function under the assumption that it is governed by
macroscopic electrostatics. These theoretical developments find their
application in Monte Carlo simulations of the $2D$ fluid of dipolar hard
spheres.
  Some preliminary numerical experiments are discussed with a special emphasis
on finite size effects, a careful study of the thermodynamic limit, and a check
of the theoretical predictions for the asymptotic behavior of the pair
correlation function.
