Many fascinating phenomena occur when a submanifold of higher codimension is evolved by its mean curvature vector. Much of the structure of hypersurface flows is absent in this more general setting e.g. embeddedness and mean-convexity fail to be preserved. Consequently, even in the simplest cases (closed curves in 3-space, surfaces in 4-space) many basic questions remain unanswered. I will describe some of these, and present recent developments concerning singularity formation from joint works with Nguyen and Bourni--Langford.