  The one-loop effective action for a generic set of quantum fields is
calculared as a nonlocal expansion in powers of the curvatures (field
strengths). This expansion is obtained to third order in the curvature. It is
stressed that the covariant vertices are finite. The trace anomaly in four
dimensions is obtained directly by varying the effective action. The nonlocal
terms in the action, producing the anomaly, contain non-trivial functions of
three operator arguments. The trace anomaly is derived also by making the
conformal transformation in the heat kernel.
