  We investigate the superconformal transformation properties of Green
functions with one or more insertions of the supercurrent in N=1 supersymmetric
quantum field theories. These Green functions are conveniently obtained by
coupling the supercurrent and its trace to a classical supergravity background.
We derive flat space superconformal Ward identities from diffeomorphisms and
Weyl transformations on curved superspace. For the classification of potential
quantum superconformal anomalies in the massless Wess-Zumino model on curved
superspace a perturbative approach is pursued, using the BPHZ scheme for
renormalisation. By deriving a local Callan-Symanzik equation the usual
dilatational anomalies are identified and it is shown that no further
superconformal anomalies involving the dynamical fields are present.
