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This paper establishes the relation between traditional results from(euclidean) twistor theory and chiral formulations of General Relativity (GR),especially the pure connection formulation. Starting from a $SU(2)$-connectiononly we show how to construct natural complex data on twistor space, mainly analmost Hermitian structure and a connection on some complex line bundle. Onlywhen this almost Hermitian structure is integrable is the connection related toan anti-self-dual-Einstein metric and makes contact with the usual results.This leads to a new proof of the non-linear-graviton theorem. Finally wediscuss what new strategies this "connection approach" to twistors suggests forconstructing a twistor action for gravity. In appendix we also review all knownchiral Lagrangians for GR.

Pure Connection Formulation, Twistors and the Chase for a Twistor Action for General Relativity