This paper proposes a class of quasi-ARMAX models for non-linear systems. Similar to ordinary non-linear ARMAX models, the quasi-ARMAX models are flexible black-box models, but they have various linearity properties similar to those of linear ARMAX models. A modelling scheme is introduced to construct models consisting of two parts: a macro-part and a kernel-part. By using Taylor expansion and other mathematical transformation techniques, it is first constructed as a class of quasi-ARMAX interfaces (macro-parts) that have various linearity properties but contain some complicated coefficients. MIMO neurofuzzy models (kernel-parts) are then introduced to represent the complicated coefficients. It is shown that the proposed quasi-ARMAX models have both good approximation ability and some easy-to-use properties. The proposed models have been successfully applied to prediction, fault detection and adaptive control of non-linear systems.