This article considers the numerical treatment of differential-algebraic systems by implicit Runge-Kutta methods. The perturbation index of a problem is discussed and its relation to the numerical solution is explained. Optimal convergence results of implicit Runge-Kutta methods for problems of index 1, 2, and 3 in Hessenberg form are then surveyed and completed. Their importance in the study of convergence for singular perturbation problems is shown and some comments on the numerical treatment of stiff Hamiltonian systems are given.