This volume incorporates research papers and surveys written by participants to the International Scientific Programme on Approximation Theory, jointly supervised by the Institute for Constructive Mathematics of the University of South Florida at Tampa, USA, and the Euler International Mathematical Institute of St Petersburg, Russia. The aim of the programme was to present new developments in constructive approximation theory. The contributors discussed such topics as: the asymptotic behaviour of orthogonal polynomials; the rational approximation of classical functions; quadrature formulas; the theory of n-widths; nonlinear approximation in Hardy algebras; numerical results on best polynomial approximations; and wavelet analysis.