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.