Abstract: Many practical applications in such diverse sciences as geophysics, computer geometry, and astronomy require the modeling of an unknown "target" function on a Euclidean sphere, based on the values of the function at a finite subset of the sphere. Moreover, there is no control on the placement of the sites where the function is sampled. We provide an introduction to recent progress in this direction, mostly that made by this author and his collaborators. In particular, the quasi-interpolatory linear polynomial operators discussed here seem to give a far superior performance than traditional techniques of interpolation and least squares.