The importance of the concept of persistent excitation (PE) in adaptive identification and control has been recognized for some time. Recently it has become evident that it also plays a central role in many questions related to the robustness of adaptive systems. There is every reason to believe that arguments involving this concept will continue to feature prominently in the analysis of most of the important problems of adaptive control. Hence there is a real need for a deeper understanding of the concept. The paper is written with three objectives. The first, which is tutorial in nature, is to provide a general framework for the discussion of persistent excitation and to collect results in the area, which are scattered throughout the adaptive literature. The second objective is to present some new results related to the uniform asymptotic stability (u.a.s.) and robustness of adaptive systems and the relation of PE to the stability properties of a class of non-linear systems. The final objective is to discuss the implications of PE in future problems, including adaptive control using a reduced-order model.