The most profound connections between twistor theory and general relativity arise in connection with the relation between curved twistor spaces and curved space-time. In Penrose, the very first paper written on twistor theory, there is an interesting section on the metrical structure of Minkowski space, where the concept of what we now call the infinity twistor is introduced, and it is indicated in passing that by use of a different type of infinity twistor it is possible to represent within the twistor framework global properties of the de Sitter space-time and the Einstein static universe. A connection with flat space twistors arises from the observation due to L.Witten that the vacuum equations for space-times with two commuting Killing vectors can effectively be reduced to the self-dual Yang-Mills equations with a two-dimensional symmetry group.