In this study, we introduce and investigate a class of neural architectures of polynomial neural networks (PNNs), develop a comprehensive design methodology and carry out a series of numeric experiments. PNN is a exible neural architecture whose structure (topology) is developed through learning. In particular, the number of layers of the PNN is not ÿxed in advance but is generated on the y. In this sense, PNN is a self-organizing network. We distinguish between two kinds of SOPNN architectures, that is, (a) polynomial neuron (PN) based and (b) fuzzy polynomial neuron (FPN) based self-organizing networks. This taxonomy is based on the character of each neuron structure in the network. Each of them comes with two structures referred here to as basic and the modiÿed topology. Moreover, for each topology of the SOPNN we identify two types that is a generic and advanced type. The essence of the design procedure of self-organizing polynomial neural networks (SOPNN) dwells on the group method of data handling (GMDH) (IEEE Trans. Systems Man and Cybernet. 12 (1971) 364). Each node of the PN based SOPNN exhibits a high level of exibility and realizes a polynomial type of mapping (linear, quadratic, and cubic) between input and output variables. FPN based SOPNN dwells on the ideas of fuzzy rule-based computing and neural networks. Especially in FPN based SOPNN, the generic rules in the system assume the form "if A then y = P(x)" where A is a fuzzy relation in the condition space while P(x) is a polynomial forming a conclusion part of the rule. Each FPN (processing element) consists of a series of the nonlinear inference rules. The conclusion part of the rules, especially the regression polynomial uses several types of high-order polynomials such as linear, quadratic, modiÿed quadratic, and cubic. As the premise part of the rules, both triangular and Gaussian-like