The identification (and control) of nonlinear systems is one of the most important and actual research directions. Moreover, many systems are multivariable. Different from linear system identification (where only a few classes of models are available), in the case of nonlinear systems, the class set of models is quite diverse. One of the most appealing nonlinear models belongs to the nonlinear ARMAX (NARMAX) class. This article focusses on the identification of such a model, which can be compared with other models (such as nonlinear ARX (NARX) and linear ARMAX) in an application based on the didactical installation ASTANK2. The mathematical foundation of NARMAX models and their identification method are described at length within this article. One of the most interesting parts is concerned with the identification of optimal models not only in terms of numerical parameters but also as structure. A metaheuristic (namely, the Cuckoo Search Algorithm) is employed with the aim of finding the optimal structural indices based on a special cost function, referred to as fitness. In the end, the performances of all three models (NARMAX, NARX, and ARMAX) are compared after the identification of the ASTANK2 installation.