Let X be the Sobolev space of functions F on Rn whose m-th derivatives belong to Lp. Given a real-valued function f defined on a (given) arbitrary set E ⊂ Rn, we ask: How can we decide whether f can be extended to a function F belonging to X? If such an F exists, then how small can we take its norm in X? What can we say about F and its derivatives at a given point? Can we take an F to depend linearly on f while keeping the order of magnitude of the Sobolev norm of F as small as possible? If E is finite, can we compute an F whose Sobolev norm is of least possible order of magnitude? If so, how many computer operations does it take? The analogues of these questions for X = C m(Rn) are rather well understood. This mini-course explains what we know when X is a Sobolev space, sketches the proofs of some of the main results, and works out a few examples.