  We define, on smooth manifolds, the notions of almost twistorial structure
and twistorial map, thus providing a unified framework for all known examples
of twistor spaces. The condition of being harmonic morphisms naturally appears
among the geometric properties of submersive twistorial maps between
low-dimensional Weyl spaces endowed with a nonintegrable almost twistorial
structure due to Eells and Salamon. This leads to the twistorial
characterisation of harmonic morphisms between Weyl spaces of dimensions four
and three. Also, we give a thorough description of the twistorial maps with
one-dimensional fibres from four-dimensional Weyl spaces endowed with the
almost twistorial structure of Eells and Salamon.
