In its original form, twistor theory encodes physical fields on Minkowski space into complex analytic objects on twistor space via the Penrose transform. This is especially natural for massless fields of arbitrary spin. In the first instance these are obtained via contour integral formulae in terms of free holomorphic functions on regions in twistor space. The holomorphic twistor functions that give rise to solutions to the massless field equations are more correctly understood as Čech representatives of analytic cohomology classes on regions in formula_1. These correspondences have been extended to certain nonlinear fields, including self-dual gravity in Penrose's nonlinear graviton construction and self-dual Yang–Mills fields in Ward construction; the former gives rise to deformations of the underlying complex structure of regions in formula_1, and the latter to certain holomorphic vector bundles over regions in formula_1. These constructions have had wide applications.