Delsarte's method provides a framework for analysis of sphere packing problems in terms of positive definite functions. Constrained packing, where only a pre-specified set of angles between points may occur, is, in turn, naturally governed by the interpolation theory for positive definite functions. In this talk, we discuss the general theory of positive definite functions on the sphere, Delsarte's method and its geometric kernel embedding interpretation, related interpolation problems, and other topics. Based on joining work with Sujit Sakharam Damase.