I will describe joint work with Matt Elsey and Peter Smereka on large scale simulations of mean curvature and related geometric motions for networks of curves in 2D and surfaces in 3D. These motions arise as gradient flow for variational models encountered in image processing (Mumford-Shah model of image segmentation) and materials science (grain boundary motion in polycrystals). Our methods allow simulations with hundreds of thousands of regions, both in 2D and 3D, on uniform grids, achieve good accuracy, and require modest hardware.