  We present a general method of deriving the effective action for conformal
anomalies in any even dimension, which satisfies the Wess-Zumino consistency
condition by construction. The method relies on defining the coboundary
operator of the local Weyl group, and giving a cohomological interpretation to
counterterms in the effective action in dimensional regularization with respect
to this group. Non-trivial cocycles of the Weyl group arise from local
functionals that are Weyl invariant in and only in the physical even integer
dimension. In the physical dimension the non-trivial cocycles generate
covariant non-local action functionals characterized by sensitivity to global
Weyl rescalings. The non-local action so obtained is unique up to the addition
of trivial cocycles and Weyl invariant terms, both of which are insensitive to
global Weyl rescalings. These distinct behaviors under rigid dilations can be
used to distinguish between infrared relevant and irrelevant operators in a
generally covariant manner. Variation of the $d=4$ non-local effective action
yields two new conserved geometric stress tensors with local traces. The method
may be extended to any even dimension by making use of the general construction
of conformal invariants given by Fefferman and Graham. As a corollary,
conformal field theory behavior of correlators at the asymptotic infinity of
either anti-de Sitter or de Sitter spacetimes follows, i.e. AdS$_{d+1}$ or
deS$_{d+1}$/CFT$_d$ correspondence. The same construction naturally selects all
infrared relevant terms (and only those terms) in the low energy effective
action of gravity in any even integer dimension. The infrared relevant terms
arising from the known anomalies in d=4 imply that the classical Einstein
theory is modified at large distances.
