Abstract: Despite a long history and many results on motion by mean curvature and homogenization, there are still many interesting questions when these two problems are put together. This is a typical question of gradient flow on wiggly energy landscapes. The talk will revisit the problem of motion by mean curvature in heterogeneous medium, reviewing some earlier and recent results. It will also present an evolutionary variational approach to analyze time dependent homogenization in L^2 and Wasserstein spaces.