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,

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... by 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.