In the first part of the talk (joint work with M. Neal), we give some new characterizations, of unitaries, isometries, unital operator spaces, unital function spaces, function systems, operator systems, C*-algebras, and related objects. In the second part (joint work with B. Magajna), we investigate the duality of operator systems and unital operator spaces. In particular, we characterize weak* closed unital operator spaces and systems, and dual function systems.