We shall discuss a recent result on uniform convergence of metrics $g_k$ on a closed surface with bounded integral curvature (measure) in the sense of A.D. Alexandrov, under the assumption that the curvature measures $K_{g_k}=\mu^1_k-\mu^2_k$, where $\mu^1_k,\mu^2_k$ are Radon measures that converge weakly to measures $\mu^1,\mu^2$ respectively, and $\mu^1$ is less than $2\pi$ at each point. This gives a global version of Yu. G. Reshetnyak’s well-known result on uniform convergence of such metrics on a domain in $C$, and answers affirmatively the open question about the metric convergence on a closed surface. This is joint work with Yuxiang Li.