A smoothing and extrapolation method is applied to the point kinetics equations and the one-dimensional space-dependent reactor kinetics equations. The simple smoothing procedure is shown to be very efficient in reducing the oscillatory errors that occur when the standard Padé(1,1) and Crank-Nicholson approximations are applied to stiff reactor kinetics equations. Fourth-order accuracy is achieved by applying a single Richardson extrapolation (on a global basis) to the smoothed results obtained from values calculated using two time-step grids. The numerical results for point kinetics demonstrate that the method is particularly efficient for very stiff problems such as subcritical and delayed supercritical transients in fast reactors. Application of the method to two one-dimensional kinetics benchmark problems solved using a standard space-dependent computer code that utilizes the Crank-Nicholson approximation leads to significant reduction in the overall computational effort required to achieve a given accuracy.