System identification is a mature field in physical sciences and an emerging field in social sciences, with a vast range of applications. Nevertheless, it remains of great focus in academia. The main challenge is the efficient use of data to generate good model fits. System identification involves multi-disciplinary techniques from statistical, mathematical and computational sciences. The typical approaches for dynamic system identification include fuzzy models, non-linear auto regressive models, state-space models, subspace identification models and many others. In this thesis, artificial neural networks are evaluated, among these, as black-box methods known to be capable of universal approximation. With no essential prior information, the identification problem exhibits more difficult challenges. These include the complexity of the resulting models, choice of regressors, and uncertainty quantification. Specifically in this thesis, a Sparse Bayesian Learning approach is proposed, as a solution to these challenges. A practical iterative Bayesian procedure is derived and set to identify six benchmark datasets of three non-linear mechanical processes: Cascaded Tanks, Coupled Electric Drives, Bouc-Wen hysteresis model as well as of three linear mechanical processes: Heat Exchanger, Glass Tube Manufacturing and Hair Dryer.