abstract: How can we tell whether a given real-valued function f on an arbitrary given subset of Rn extends to a Cm function F on the whole Rn? If F exists, how small can we take its Cm norm? What can we say about the derivatives of F at a given point? Can we make F depend linearly on f? Suppose f is defined only on a finite set. How can we compute an extension F with close-to-minimal norm? How many computer operations does it take? What if F is required merely to agree approximately with f? What if we are allowed to discard a few of the points of E? What about function spaces other than Cm? Many of the results discussed are joint work with Bo'az Klartag.