We will investigate the use of weighted triangulations as a discrete analogue of Riemannian geometry. We will then introduce discrete Laplacian operators, which are particularly weighted Laplacians on the 1-skeleton of a metric (Euclidean) triangulation in the sense of Laplacians on graphs. We will investigate some of the properties of these Laplacians, including an interesting optimality result for weighted Delaunay triangulations originally proven by Rippa for (unweighted) Delaunay triangulations.