Many relativistic quantum field theories have interacting renormalization group fixed points, which have scale symmetry in addition to Poincare symmetry. In almost all cases one finds that the spacetime symmetry is enhanced to conformal invariance, and further to Weyl invariance. This talk reports on progress in understanding these symmetry enhancements in spacetime dimensions greater than 2. I will present a sufficient condition for conformal invariance that applies to many interesting renormalization group fixed points, including the critical point of the 3D Ising model. I also demonstrate that conformal invariance implies Weyl invariance for unitary theories.