We describe numerical methods for the construction of interpolatory quadrature rules of Radau and Lobatto types. In particular, we are interested in deriving efficient algorithms for computing optimal averaged Gauss–Radau and Gauss–Lobatto type javascript:undefined;quadrature rules. These averaged rules allow us to estimate the quadrature error in Gauss–Radau and Gauss–Lobatto quadrature rules. This is important since the latter rules have higher algebraic degree of exactness than the corresponding Gauss rules, and this makes it possible to construct averaged quadrature rules of higher algebraic degree of exactness than the corresponding “standard” averaged Gauss rules available in the literature.