A criterion is proposed to ensure that classical relativistic fields do not propagate superluminally. If this criterion does indeed serve as a sufficient condition for no superluminal propagation it follows that various other criteria found in the physics literature cannot serve as necessary conditions since they can fail although the proffered condition holds. The rejected criteria rely on energy conditions that are believed to hold for most classical fields used in actual applications. But these energy conditions are known to fail (...) at small scales for quantum fields. It is argued that such a failure is not necessarily a cause for concern about superluminal propagation in the quantum regime since the proffered criterion of no superluminal propagation for classical fields has a natural analog for quantum fields and, further, this quantum analog condition provably holds for some quantum fields despite the violation of energy conditions. The apparatus developed here also offers a different approach to treating the Reichenbach-Salmon cases of "pseudo-causal processes" and helps to clarify the issue of whether relativity theory is consistent with superluminal propagation. (shrink)