Suppose that X is a metric space and that x_1,...,x_n are n points in X. Every 1-Lipschitz function from {x_1,...,x_n} to a normed space Z can be extended to a Z-valued function that is defined on all of X and has Lipschitz constant that is bounded from above by a quantity that depends only on n. It is a longstanding question to determine the best possible asymptotic dependence on n here. This talk will start by explaining the long line of works on this question starting from the early 1980s, involving a variety of analytic, probabilistic, combinatorial and geometric techniques, followed by a description of recent progress.