  We develop a non-relativistic twistor theory, in which Newton--Cartan
structures of Newtonian gravity correspond to complex three-manifolds with a
four-parameter family of rational curves with normal bundle ${\mathcal
O}\oplus{\mathcal O}(2)$. We show that the Newton--Cartan space-times are
unstable under the general Kodaira deformation of the twistor complex
structure. The Newton--Cartan connections can nevertheless be reconstructed
from Merkulov's generalisation of the Kodaira map augmented by a choice of a
holomorphic line bundle over the twistor space trivial on twistor lines. The
Coriolis force may be incorporated by holomorphic vector bundles, which in
general are non--trivial on twistor lines. The resulting geometries agree with
non--relativistic limits of anti-self-dual gravitational instantons.
