For each point x in R^n, suppose we're given a convex set K(x) in R^D. For fixed m, we want to find a C^m function F:R^n->R^D such that F(x) lies in K(x) for each x. Under mild restrictions on the family K(x), a recent result of Jiang, Luli and O'Neill gives necessary and sufficient conditions for the existence of such an F. The course starts with elementary background and builds up to the proof of that result. No special background is assumed.