  We consider a general class of four-dimensional geometries admitting a null
vector field that has no twist and no shear but has an arbitrary expansion. We
explicitly present the Petrov classification of such Robinson-Trautman (and
Kundt) gravitational fields, based on the algebraic properties of the Weyl
tensor. In particular, we determine all algebraically special subcases when the
optically privileged null vector field is a multiple principal null direction
(PND), as well as all the cases when it remains a single PND. No field
equations are a priori applied, so that our classification scheme can be used
in any metric theory of gravity in four dimensions. In the classic Einstein
theory this reproduces previous results for vacuum spacetimes, possibly with a
cosmological constant, pure radiation and electromagnetic field, but can be
applied to an arbitrary matter content. As non-trivial explicit examples we
investigate specific algebraic properties of the Robinson-Trautman spacetimes
with a free scalar field, and also black hole spacetimes in the pure
Einstein-Weyl gravity.
