  We reformulate Euclidean general relativity without cosmological constant as
an action governing the complex structure of twistor space. Extending Penrose's
non-linear graviton construction, we find a correspondence between twistor
spaces with partially integrable almost complex structures and four-dimensional
space-times with off-shell metrics. Using this, we prove that our twistor
action reduces to Plebanski's action for general relativity via the Penrose
transform. This should lead to new insights into the geometry of graviton
scattering as well as to the derivation of computational tools like
gravitational MHV rules.
