Offers an accessible, self-contained collection of work by researchers at the frontier of developments in the field, most of which is unavailable elsewhere in the literatureDocuments the development of the original nonlinear graviton construction and its extensions and applications from the perspective of differential geometryPresents different approaches to finding a twistor correspondence for space-times in four dimensions that are not necessarily anti-self-dualExplores hypersurface twistor spacesExamines various attempts to find a more fundamental twistor correspondence for vacuum space-times Although twistor theory originated as an approach to the unification of quantum theory and general relativity, twistor correspondences and their generalizations have provided powerful mathematical tools for studying problems in differential geometry, nonlinear equations, and representation theory. At the same time, the theory continues to offer promising new insights into the nature of quantum theory and gravitation.Further Advances in Twistor Theory, Volume III: Curved Twistor Spaces is actually the fourth in a series of books compiling articles from Twistor Newsletter-a somewhat informal journal published periodically by the Oxford research group of Roger Penrose. Motivated both by questions in differential geometry and by the quest to find a twistor correspondence for general Ricci-flat space times, this volume explores deformed twistor spaces and their applications. Articles from the world's leading researchers in this field-including Roger Penrose-have been written in an informal, easy-to-read style and arranged in four chapters, each supplemented by a detailed introduction. Collectively, they trace the development of the twistor programme over the last 20 years and provide an overview of its recent advances and current status.