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This paper outlines how it is possible to decompose a complex non-linear modelling problem into a set of simpler linear modelling problems. Local ARMAX models valid within certain operating regimes are interpolated to construct a global NARMAX (non-linear NARMAX) model. Knowledge of the system behaviour in terms of operating regimes is the primary basis for building such models, hence it should not be considered as a pure black-box approach, but as an approach that utilizes a limited amount of a priori system knowledge. It is shown that a large class of non-linear systems can be modelled in this way, and indicated how to decompose the systems range of operation into operating regimes. Standard system identification algorithms can be used to identify the NARMAX model, and several aspects of the system identification problem are discussed and illustrated by a simulation example.