We introduce the classical extension results formulated by Whitney and later by Brudnyi-Shvartsman and Fefferman in the setting of the function space C^m. We present our theorems on interpolation of functions in Sobolev spaces. We then present our interpolation algorithms based on a recent constructive proof of our main theorem. We hope to explain the main obstacles that arose in converting our original non-constructive proof into a constructive one. If time permits, we will explain our work related to extension problems for bandlimited functions. This talk is based partly on joint work with C. Fefferman and G. Luli.