  We discuss the structure of nonlocal effective action generating the
conformal anomaly in classically Weyl invariant theories in curved spacetime.
By the procedure of conformal gauge fixing, selecting the metric representative
on a conformal group orbit, we split the renormalized effective action into
anomalous and Weyl invariant parts. A wide family of thus obtained anomalous
actions is shown to include two special cases of Riegert--Fradkin--Tseytlin and
Fradkin--Vilkovisky actions. Both actions are shown to be contained in the
first three orders of the curvature expansion for a generic one-loop effective
action obtained by covariant perturbation theory. The complementary Weyl
invariant part of the action is given by the ``conformization'' of the full
effective action -- restricting its argument to the conformally invariant
representative of the orbit of the conformal group. This is likely to resolve a
long-standing debate between the proponents of the Riegert action and adherents
of the perturbation expansion for the effective action with typical nonlocal
logarithmic form factors. We derive the relation between quantum stress tensors
on conformally related metric backgrounds, which generalizes the known
Brown-Cassidy equation to the case of nonzero Weyl tensor, and discuss
applications of this relation in the cosmological model driven by conformal
field theory. We also discuss the issue of renormalization group running for
the cosmological and gravitational coupling constants and show that it exhibits
a kind of a metamorphosis to the nonlocal form factors of the so-called
partners of the cosmological and Einstein terms -- nonlocal curvature squared
terms of the effective action.
