  We discuss conformally covariant differential operators, which under local
rescalings of the metric, \delta_\sigma g^{\mu\nu} = 2 \sigma g^{\mu\nu},
transform according to \delta_\sigma \Delta = r \Delta \sigma + (s-r) \sigma
\Delta for some r if \Delta is s-th order. It is shown that the flat space
restrictions of their associated Green functions have forms which are strongly
constrained by flat space conformal invariance. The same applies to the
variation of the Green functions with respect to the metric. The general
results are illustrated by finding the flat space Green function and also its
first variation for previously found second order conformal differential
operators acting on $k$-forms in general dimensions. Furthermore we construct a
new second order conformally covariant operator acting on rank four tensors
with the symmetries of the Weyl tensor whose Green function is similarly
discussed. We also consider fourth order operators, in particular a fourth
order operator acting on scalars in arbitrary dimension, which has a Green
function with the expected properties.
  The results obtained here for conformally covariant differential operators
are generalisations of standard results for the two dimensional Laplacian on
curved space and its associated Green function which is used in the Polyakov
effective gravitational action. It is hoped that they may have similar
applications in higher dimensions.
