  We construct the twistor space associated with an HKT manifold, that is, a
hyper-K\"ahler manifold with torsion, a type of geometry that arises as the
target space geometry in two-dimensional sigma models with (4,0) supersymmetry.
We show that this twistor space has a natural complex structure and is a
holomorphic fibre bundle over the complex projective line with fibre the
associated HKT manifold. We also show how the metric and torsion of the HKT
manifold can be determined from data on the twistor space by a reconstruction
theorem. We give a geometric description of the sigma model (4,0) superfields
as holomorphic maps (suitably understood) from a twistorial extension of (4,0)
superspace (harmonic superspace) into the twistor space of the sigma model
target manifold and write an action for the sigma model in terms of these (4,0)
superfields.
