The purpose of this book, written by world-renowned authorities in this field, is to guide the reader in learning about new developments in approximation theory that have come up over the last 20 years. The emphasis is on multivariate approximation theory. Most of the topics appear for the first time here in a systematic textbook and many are still current areas of research. The authors cover positive definite functions, radial basis interpolation, thin-plate splines, neural networks, ridge functions, box splines, approximation on spheres, and wavelets. Every chapter includes practical problems from current research in areas of science, engineering, geophysics, business, and economics.