The classical uniformization theorem transforms the study of moduli spaces of marked Riemann surfaces into the study of constant curvature metrics with singularities. I will give a survey on constant curvature metrics with cusp and conical singularities, including works joint with Richard Melrose, Rafe Mazzeo and Bin Xu, where new analytic tools have been developed to understand the uniformization. ``Resolution of singularities'' is the key idea in the analysis, which can be seen as an analogue of the Deligne--Mumford compactification of Riemann moduli spaces.