  We investigate the geometry of almost Robinson manifolds, Lorentzian
analogues of Hermitian manifolds, defined by Nurowski and Trautman as
Lorentzian manifolds of even dimension equipped with a totally null complex
distribution of maximal rank. Associated to such a structure, there is a
congruence of null curves, which, in dimension four, is geodesic and
non-shearing if and only if the complex distribution is involutive. Under
suitable conditions, the distribution gives rise to an almost Cauchy--Riemann
structure on the leaf space of the congruence.
  We give a comprehensive classification of such manifolds on the basis of
their intrinsic torsion. This includes an investigation of the relation between
an almost Robinson structure and the geometric properties of the leaf space of
its congruence. We also obtain conformally invariant properties of such a
structure, and we finally study an analogue of so-called generalised optical
geometries as introduced by Robinson and Trautman.
