  We give a complete geometric description of conformal anomalies in arbitrary,
(necessarily even) dimension. They fall into two distinct classes: the first,
based on Weyl invariants that vanish at integer dimensions, arises from finite
-- and hence scale-free -- contributions to the effective gravitational action
through a mechanism analogous to that of the (gauge field) chiral anomaly. Like
the latter, it is unique and proportional to a topological term, the Euler
density of the dimension, thereby preserving scale invariance. The
contributions of the second class, requiring introduction of a scale through
regularization, are correlated to all local conformal scalar polynomials
involving powers of the Weyl tensor and its derivatives; their number increases
rapidly with dimension. Explicit illustrations in dimensions 2, 4 and 6 are
provided.
