This centenary history of Runge-Kutta methods contains an appreciation of the early work of Runge, Heun, Kutta, and Nystr6m and a survey of some significant developments of these methods over the last hundred years. In particular, the order conditions, as they are now understood, will be outlined, as will the introduction and practical implementation of implicit Runge-Kutta methods, the use of linear and nonlinear stability analysis in the assessment of Runge-Kutta methods, and the theory and applications of the composition of methods. Of the many further developments that have arisen, most are discussed only briefly; however, the recent interest in Runge-Kutta methods for Hamiltonian systems will be surveyed in more detail.