This comprehensive treatise proposes a unified approach, based on the interplay between on optimal designs for interpolation and extrapolation and the theory of the uniform approximation of functions. This self-contained book is intended for statisticians interested in the mathematical basis of optimal planning and for mathematicians interested in the applications of approximation theory. It covers topics developed over a century, unifies concepts and notation, and explores the main results and notions of this area of statistical inference; it states explicit methods for the construction of these designs.