Let $\Sigma$ be an oriented surface. A maximal atlas on $\Sigma$ consisting of charts with biholomorphic transition functions is called a complex structure on $\Sigma$. Two metrics $g_1$ and $g_2$ on $\Sigma$ are called conformal if there exists a positive smooth function $f$ such that $g_1=f g_2$. One can define an equivalence relation on the collection of metrics on $\Sigma$ via this. Now an equivalence class on $\Sigma$ is called a conformal structure on $\Sigma$. I will try to establish a relation between these two structures and sketch a proof.