  A Riemannian or pseudo-Riemannian (or conformal) structure is conformally
Einstein if and only if there is a suitably generic parallel section of a
certain vector bundle -- the so-called standard conformal tractor bundle. We
show that this characterisation leads to a systematic approach to constructing
obstructions to conformally Einstein metrics. Relaxing the requirement that the
parallel tractor field be generic gives a natural generalisation of the
Einstein equations.
