Discrete conformal mappings based on circle packing, vertex scaling, and related structures has had significant activity since Thurston proposed circle packing as a way to approximate conformal maps in the 1980s. The first convergence result of Rodin-Sullivan (1987) proved that circle packing maps do indeed converge to conformal maps to the disk. Recent results have shown convergence of maps of other discrete conformal structures to conformal maps as well (e.g., Luo-Sun-Wu 2022). In this talk, we will discuss a general method to approach convergence of discrete conformal mappings between surfaces. It is notable that this convergence result does not rely on a uniformization. Instead, direct mappings from a piecewise Euclidean space to the Riemannian surface is given using Riemannian barycentric coordinates using Karcher’s center of mass technique. Estimates on the discrete conformal structure lead to a comparison between the pullback of a Riemannian metric under a conformal map and pullback of the metric under the composition of barycentric coordinates and discrete conformal maps.