  We find that the target space of two-dimensional (4,0) supersymmetric sigma
models with torsion coupled to (4,0) supergravity is a QKT manifold, that is, a
quaternionic K\"ahler manifold with torsion. We give four examples of
geodesically complete QKT manifolds one of which is a generalisation of the
LeBrun geometry. We then construct the twistor space associated with a QKT
manifold and show that under certain conditions it is a K\"ahler manifold with
a complex contact structure. We also show that, for every 4k-dimensional QKT
manifold, there is an associated 4(k+1)-dimensional hyper-K\"ahler one.
