摘要：Singular spherical,flat and hyperbolic metrics are conformal metrics with constant curvature +1, 0 and ?1, respectively, and with isolated singularities on Riemann surfaces. The Gauss-Bonnet formula gives a necessary condition for the existence of such three kinds of metrics with prescribed conical singularities on compact Riemann surfaces,and it is also sufficient for both cone flat and cone hyperbolic metrics. However, it is not the case for cone spherical metrics, whose existence has been an open problem over twenty years on compact Riemann surfaces. I will introduce the respectful audience some progress on this problem and some recent results on singular flat metrics and hyperbolic metrics ones. The talk is based my joint works with Qing Chen, Xuemiao Chen, Yiran Cheng, Yu Feng, Si-en Gong, Bo Li, Jin Li, Lingguang Li, Hongyi Liu, Santai Qu, Jijian Song, Yingyi Wu, Xuwen Zhu.