We will discuss some results on the motion of interfaces by its mean curvature in heterogeneous media, in both continuous and discrete environments. Motion by mean curvature is a prototype of geometric motion which is a gradient flow (of the surface energy). Heterogeneity leads to wiggling energy landscape. Their combination gives rise to interesting questions about surface dynamics, pinning-depinning phenomena and the underlying pattern formation. The above can be demonstrated even in very simple spin systems with nearest and next-nearest neighbor interactions. Speaker(s): Aaron Yip (Purdue University)