By generalizing the recent proof of the a-theorem, I derive constraints on the possible UV and IR asymptotics of 4D Lorentz invariant unitary quantum field theory. Within perturbation theory the only possible RG flow asymptotic is given by conformal field theory. I also give a non-perturbative argument that excludes theories with scale but not conformal invariance. This argument holds for theories in which the stress-energy tensor is sufficiently nontrivial in a definite technical sense.