The book incorporates research papers and surveys written by participants ofan International Scientific Programme onApproximation Theory jointly supervised by Institute forConstructive Mathematics of University of South Florida atTampa, USA and the Euler International MathematicalInstituteat St. Petersburg, Russia. The aim of theProgramme was to present new developments in ConstructiveApproximation Theory. The topics of the papers are:asymptotic behaviour of orthogonal polynomials, rationalapproximation of classical functions, quadrature formulas,theory of n-widths, nonlinear approximation in Hardyalgebras,numerical results on best polynomial approximations, wavelet analysis.FROM THE CONTENTS: E.A. Rakhmanov: Strong asymptotics fororthogonal polynomials associated with exponential weightson R.- A.L. Levin, E.B. Saff: Exact Convergence Rates forBest Lp Rational Approximation to the Signum Function andfor Optimal Quadrature in Hp.- H. Stahl: Uniform RationalApproximation of x .- M. Rahman, S.K. Suslov: Classical Biorthogonal Rational Functions.- V.P. Havin, A. PresaSague: Approximation properties of harmonic vector fieldsand differential forms.- O.G. Parfenov: Extremal problemsfor Blaschke products and N-widths.- A.J. Carpenter, R.S.Varga: Some Numerical Results on Best Uniform Polynomial Approximation of x on 0,1 .- J.S. Geronimo: PolynomialsOrthogonal on the Unit Circle with Random Recurrence Coefficients.- S. Khrushchev: Parameters of orthogonal polynomials.- V.N.Temlyakov: The universality of the Fibonacci cubature formulas.