We consider the following mean field equation:
$$\Delta_{g}v+\rho\left(\frac{h^*e^v}{\int_Mh^*e^v}-1\right)=4\pi\sum_{j=1}^N\alpha_j(\delta_{q_j}-1)$$ on M,
where M is a compact Riemann surface with volume 1, $h^*$ is a positive $C^1$ function on M,
and $\alpha_j$ are constants satisfying $\alpha_j>-1$. In this paper, we derive the topological
degree counting formula for noncritical values of $\rho$. We also give several applications of this
formula, including existence of curvature +1 metric with conic singularities, doubly periodic
solutions of electroweak theory, and a special case of self-gravitating strings.