We propose a new category of neurofuzzy networks- Self-organizing Neural Networks(SONN) with fuzzy polynomial neurons(FPNs) and discuss a comprehensive design methodology supporting their development. Two kinds of SONN architectures, namely a basic SONN and a modified SONN architecture are dicussed. Each of them comes with two types such as the generic and the advanced type. SONN dwells on the ideas of fuzzy rule-based computing and neural networks. Simulation involves a series of synthetic as well as experimental data used across various neurofuzzy systems. A comparative analysis is included as well.