This work has been designed as an overview of contemporary numerical methods, particularly predictor-corrector and Runge-Kutta methods for non-stiff problems, backward differentiation formulae and implicit Runge-Kutta methods for stiff problems. Also included are several new topics such as the structure of variable-step, variable-order predictor-corrector algorithms, the Butcher theory of Runge-Kutta methods, order stars and nonlinear stability theory.