Gaussian and Chiral beta-Ensembles, which generalise well-known orthogonal (beta = 1), unitary (beta = 2), and symplectic (beta = 4) ensembles of random Hermitian matrices, are considered. Averages are shown to satisfy duality relations like {beta, N, n} double left right arrow {4/beta, n, N} for all beta > 0, where N and n respectively denote the number of eigenvalues and products of characteristic polynomials. At the edge of the spectrum, matrix integrals of the Airy (Kontsevich) type are obtained. Consequences on the integral representation of the multiple orthogonal polynomials and the partition function of the formal one-matrix model are also discussed. Proofs rely on the theory of multivariate symmetric polynomials, especially Jack polynomials. (C) 2009 Elsevier B. V. All rights reserved.