  We consider the mean field equation: (1) \Delta u+\rho\frac{e^u}{\int_\Omega
e^u}=0 & \hbox{in} \;\Omega, u=0 & \hbox{on}\;\partial\Omega, where
$\Omega\subset \mathbb{R}^2$ is an open and bounded domain of class $C^1$. In
his 1992 paper, Suzuki proved that if $\Omega$ is a simply-connected domain,
then equation (1) admits a unique solution for $\rho\in[0,8\pi)$. This result
for $\Omega$ a simply-connected domain has been extended to the case
$\rho=8\pi$ by Chang, Chen and the second author. However, the uniqueness
result for $\Omega$ a multiply-connected domain has remained a long standing
open problem which we solve positively here for $\rho\in[0,8\pi]$. To obtain
this result we need a new version of the classical Bol's inequality suitable to
be applied on multiply-connected domains.
  Our second main concern is the existence of solutions for (1) when
$\rho=8\pi$. We a obtain necessary and sufficient condition for the solvability
of the mean field equation at $\rho=8\pi$ which is expressed in terms of the
Robin's function $\gamma$ for $\Omega$. For example, if equation (1) has no
solution at $\rho=8\pi$, then $\gamma$ has a unique nondegenerate maximum
point. As a by product of our results we solve the long-standing open problem
of the equivalence of canonical and microcanonical ensembles in the Onsager's
statistical description of two-dimensional turbulence on multiply-connected
domains.
