This volume presents main results of the Fourth International Conference on Multivariate Approximation held at Witten-Bommerholz in 2000. The book starts with article by M. Reimer and H. Schwetlick: J. W. Schmidt in Memoriam. Jochen W. Schmidt (1931 - 2000) was twice selected by the mathematical community as a leading mathematics reviewer of the German science foundation and participated in the preceding Bommerholz Conferences on Multivariate Approximation. The proceedings contains nineteen selected, peer-reviewed contributions covering the following main topics: interpolation and approximation on compact sets, Kergin interpolation; error asymptotics; radial basic functions; energy minimizing configurations on the sphere; quadrature and cubature formulae, harmonic functions near a zero; blending functions; frames and approximation of inverse frame operators. The list of publications and further data on J. W. Schmidt are enclosed in an appendix.