We discuss possible algorithms for interpolating data given in a set of curves and/or points in a surface in R³. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The absolute minimal Lipschitz extension model (AMLE) is singled out and studied in more detail. We show experiments illustrating the interpolation of data on the sphere and the torus.