This volume containing the proceedings of the NATO Advanced Study Institute embodies survey articles and recent results from the theory of splines, spline-wavelets, multivariate wavelet decomposition, interpolation theory, polynomical approximation, near-minimax approximations, rational interpolants, and Pade approximants in one and several variables, radial basis approximation, optimixation theory, orthogonal polynomials, and more. The volume should therefore be of interest to researchers and graduate students in mathematics and engineering for whom the latest developments in approximation theory are of interest.