As W. Gautschi recalled in [1], it was in 1969 when the connection between Gauss quadrature rules and the algebraic eigenvalue problem was, if not discovered, certainly exploited in the now classical and widely cited paper [2]. D. P. Laurie has written that the timeless fact about the Golub-Welsch approach is that any advance in our ability to solve the symmetric tridiagonal eigenproblem implies a corresponding advance in our ability to compute Gaussian formulas. Our aim in this talk is to show how the total positivity of Jacobi matrices of some families of orthogonal polynomials, along with the accurate computation of their bidiagonal decompositions, can help to compute more accurately the nodes and the weights of the corresponding Gauss quadrature rules, by using the algorithms developed by P. Koev [3].