These notes arose from a series of lectures first presented at the Scuola Internazionale Superiore di Studi Avanzati and the International Centre for Theoretical Physics in Trieste in July 1980 and then, in an extended form, at the Universities of Sofia (1980-81) and Bielefeld (1981). Their objective has been two-fold. First, to introduce theorists with some background in group representations to the notion of twistors with an emphasis on their conformal properties; a short guide to the literature on the subject is designed to compensate in part for the incompleteness and the one-sidedness of our review. Secondly, we present a systematic study of positive energy conformal orbits in terms of twistor flag manifolds,
