  Green functions play an important role in conformal geometry. In this paper,
we explain how to compute explicitly the logarithmic singularities of the Green
functions of the conformal powers of the Laplacian. These operators include the
Yamabe and Paneitz operators, as well as the conformal fractional powers of the
Laplacian arising from scattering theory for Poincar\'e-Einstein metrics. The
results are formulated in terms of Weyl conformal invariants arising from the
ambient metric of Fefferman-Graham. As applications we obtain "Green function"
characterizations of locally conformally flat manifolds and a spectral
theoretic characterization of the conformal class of the round sphere.
