The trace anomaly in external gravity is the sum of three terms at criticality: the square of the Weyl tensor, the Euler density and square R, with coefficients, properly normalized, called c, a, and a', the latter being ambiguously defined by an additive constant. Considerations about unitarity and positivity properties of the induced actions allow us to show that the total RG flows of a and a' are equal and therefore the a'-ambiguity can be consistently removed through the identification a'=a. The picture that emerges clarifies several long-standing issues. The interplay between unitarity and renormalization implies that the flux of the renormalization group is irreversible. A monotonically decreasing a-function interpolating between the appropriate values is naturally provided by a'. The total a-flow is expressed non-perturbatively as the invariant (i.e., scheme-independent) area of the graph of the beta function between the fixed points. We test this prediction to the fourth loop order in perturbation theory, in QCD with Nf less than or similar to 1 1/2 N-c and in supersymmetric QCD. There is agreement also in the absence of an interacting fixed point (QED and phi(4)-theory). Arguments for the positivity of a are also discussed. (C) 1999 Academic Press.