This chapter is concerned with the overview that twistor theory provides over the theory of nonlinear integrable equations. The main obstacle to the twistor programme is the lack of a twistor description of the full Einstein and Yang-Mills equations in Lorentz signature. In the case of the Einstein vacuum equations, the linear system is the Rarita Schwinger equation for the potential of a helicity-3/2 field and in the case of the Yang-Mills equations, the linear system is just the linearized equations off the given background. The chapter reviews the standard correspondence between solutions of the Bogomoln ‘yi hierarchy and bundles on O(n) in which the bundle will be described by its ∂¯-operator. It provides a formulation of a construction for solutions of the Kadomtsev-Petviashvilii equations in which the ∂¯-operator of the standard twistor construction is generalized to a Dirac operator.