R. Penrose (in Advances in twistor theory, pp. 168-176, and in Cosmology and gravitation; Nato advanced study institute series, pp. 287-316 (1980). New York: Plenum Press) has argued that the goal of twistor theory with regard to the vacuum Einstein equations ought to consist of some kind of unification of twistor-theoretic descriptions of anti-self-dual (a.s.d.) and self-dual (s.d.) space-times. S.d. space-times currently possess a description only in terms of dual twistor space, however, rather than twistor space. In this paper, suggestions due to Penrose for providing a purely twistor space description of s.d. space-times are investigated. It is shown how the points of certain s.d. space-times define mappings on twistor space and the geometry of these mappings is studied. The families of mappings for two particular s.d. space-times are presented explicitly.