Gauss quadrature is a well known method for computing the definite integrals numerically and almost every textbook of numerical analysis devotes one chapter to this topic. The main advantage of Gauss quadrature is its high accuracy, i.e., a (n+1)-point Gauss quadrature integrates exactly polynomials of degree up to 2n+1. Recently, Clenshaw-Curtis quadrature, which is the interpolatory quadrature rule based on Chebyshev-Lobatto points, has received much attention due to its high efficiency and accuracy. In this talk, I shall give some recent progress on the convergence rates of Gauss and Clenshaw-Curtis quadrature rules. A comparison of both quadrature rules is given to illustrate their efficiency and accuracy for analytic, differentiable and singular functions.