We give an introduction to twistor theory and construct a twistor action for the self-dual part of conformal higher spin theories by treating the integrability condition for the holomorphic structure of a complex deformation of flat twistor space as the equations of motion of those higher spin fields. Analogous to the embedding of Einstein gravity with cosmological constant in Weyl gravity, we identify a ghost-free subsector which generates the unique three-point anti-MHV amplitude consistent with Poincaré invariance and helicity constraints.