The identification of non-linear systems based on a NARMAX (Non-linear AutoRegressive Moving-Average model with exogenous inputs) model representation is considered, and a combined stepwise-regression/prediction-error estimation algorithm is derived. The stepwise-regression routine determines the model structure by detecting significant terms in the model, while the prediction-error algorithm provides optimal estimates of the final model parameters. Implementation of the algorithms is discussed in detail, and several simulated examples and industrial applications are included to illustrate that parsimonious models of non-linear systems can be identified using the algorithm.