Abstract - The a-theorem generalizes to other dimensions a property of 2D quantum field theories, namely that there exists a function of the couplings A, such that A decreases monotonically along RG flow, and at fixed points A=a/4, the coefficient of the Euler density in the trace anomaly. We investigate consequences of this generalization in D=4, where the result holds perturbatively; D=6, where A instead increases along RG flow; and D=3, where there is seemingly no natural quantity with which we may identify A.