Abstract: In this talk, we consider the self-dual equations arising from the Einstein-Maxwell-Higgs model on compact surfaces. According to the behaviors of solutions as the coupling parameter $\epsilon$ decreases to zero, we classify solutions into two categories: topological solutions and nontopological solutions. In this talk, we review recent advances for the exsitence of solutions for both types. First, we show that there exists $\epsilon_0$ such that there are at least two solutions $v_{1,\epsilon}$ and $v_{2,\epsilon}$ for $\epsilon \in (0,\epsilon_0)$ and no solutions for $\epsilon > \epsilon_0$. It turns out that $v_{1,\epsilon}$ is topological for small $\epsilon$. Second, we present the existence of nontopological solutions for all small $\epsilon$ via a degree argument.