Twistor theory is the remarkable mathematical framework that was discovered by Roger Penrose in the course of research into gravitation and quantum theory. It have since developed into a broad, many-faceted programme that attempts to resolve basic problems in physics by encoding the structure of physical fields and indeed space-time itself into the complex analytic geometry of twistor space. Twistor theory has important applications in diverse areas of mathematics and mathematical physics. These include powerful techniques for the solution of nonlinear equations, in particular the self-duality equations both for the Yang-Mills and the Einstein equations, new approaches to the representation theory of Lie groups, and the quasi-local definition of mass in general relativity, to name but a few. This volume and its companions comprise an abundance of new material, including an extensive collection of Twistor Newsletter articles written over a period of 15 years. These trace the development of the twistor programme and its applications over that period and offer an overview on the current status of various aspects of that programme. The articles have been written in an informal and easy-to-read style and have been arranged by the editors into chapter supplemented by detailed introductions, making each volume self-contained and accessible to graduate students and non-specialists from other fields. Volume II explores applications of flat twistor space to nonlinear problems. It contains articles on integrable or soluble nonlinear equations, conformal differential geometry, various aspects of general relativity, and the development of Penrose's quasi-local mass construction.