The study of symplectic structures has its origins in classical mechanics and has seen a remarkable growth over the last quarter century. The pursuit of what might be considered the first basic question in the field - which manifolds admit symplectic structures and to what extent they are unique - has, on the one hand, led to many results based on a variety of diverse methods, and, on the other hand, left many questions wide open. In this lecture I will present a few highlights and describe some new and old conjectures in the subject.