In recent decades, twistor theory has grown into an irreplaceable tool for the study of scattering amplitudes in gauge theory and gravity. This book introduces the reader to cutting-edge advances in twistor theory and its applications to general relativity. The problem of graviton scattering in four dimensions is shown to be dual to dramatically simpler computations in a two-dimensional CFT known as a twistor sigma model. Twistor sigma models are the first step toward a holographic description of gravity in asymptotically flat space-times. They underpin the infinitely many asymptotic symmetries of flat space physics discovered in celestial holography, and extend them to exciting new arenas like curved space-times. They also yield intrinsically mathematical results in the field of hyperkähler manifolds. This volume will be of broad interest to students and researchers looking for an accessible entry point into twistor geometry, scattering amplitudes, and celestial holography. It will also provide an invaluable reference for specialists by bringing together results from a host of different disciplines.