These proceedings contain the main topics and results presented at the Fourth International Conference on Multivariate Approximation. The meeting took place during the week of September 24-29, 2000 at the now well-known 'Haus Bommerholz', the guest-house of the University of Dortmund. It hosted 43 participants from 16 countries, and the program included 9 invited one-hour lectures, 21 contributed talks, and two problem sessions. The articles collected here are carefully peer-refereed and suitably edited for publication. Following the tradition of this series of conferences, the meeting was aimed at advancing selected topics of Multivariate Approximation Theory. These include approximation on compact sets (such as spheres, balls, or compact ho mogeneous manifolds), spherical designs and energy functionals, interpolation by radial basis functions and by splines, frame theory and Gabor analysis, re finable function systems and subdivision, properties of harmonic, polyharmonic and blending functions, sampling and data compression, among others. The editors would like to express their thanks to all who have given their sup port to the conference, and to the preparation of this book. In particular, our thanks go to the Deutsche Forschungsgemeinschaft for their funding of the con ference, to the University of Dortmund, and to the staff of Haus Bommerholz for their help running the conference, and to the Universities of Dortmund and Duisburg for the financial support of this proceedings volume.