The Lipschitz extension problem asks for geometric conditions on a pair of metric spaces X and Y implying that there exists a positive constant K such that for every subset A of X, every L-Lipschitz function f from A to Y can be extended to a (KL)-Lipschitz function defined on all of X. When Y is the real line then this is always possible with K=1 (the nonlinear Hahn-Banach theorem), in which case one asks for an extension of f with additional desirable properties. For general metric spaces X,Y it is usually the case that no such K exists. However, many deep investigations over the past century have revealed that in important special cases the Lipschitz extension problem does have a positive answer. Proofs of such theorems involve methods from a variety of mathematical disciplines, and when available, a positive solution to the Lipschitz extension problem often has powerful applications. The first talk will be an introduction intended for non-experts, giving an overview of the known Lipschitz extension theorems, and an example or two of the varied methods with which such theorems are proved. The following two lectures will deal with more specialized topics, including the use of probabilistic methods, some illuminating counterexamples, examples of applications, and basic problems that remain open.