In 2-dim it is known that a unitary, well defined quantum field theory, if scale invariant must also be invariant under conformal transformations. Whether this is also true in dimensions higher than two has been an open question for decades. We have discovered renomalization group flows in 4-epsilon dimensions corresponding to scale but not conformal invariant theories. The flows correspond to limit cycles or ergodic behavior, neither of which had been reported in relativistic quantum field theories either. There seems to be a deep connection between scale but not conformal invariance and this type of renormalization group behavior. We will present these results and list some of open questions, including the possibility of such behavior in integral dimensions. |