The phenomenon of tunneling in which a quantummechanical particle can penetrate a repulsive barrier with a height that exceeds the total energy of the particle is counterintuitive. Any explanation or intuition for this process based on classical mechanics fails. At the same time, this effect is extremely important and has been studied widely. The Josephson effect in high-speed semiconductors @1#, b decay in nuclear physics, and instantons in high-energy physics are just a few examples. In the early 1930s it was already recognized that there was no appreciable temporal delay in the transmission of wave packets through barriers @2#. Wigner discussed the possibility that a particle can effectively travel faster than the speed of light when passing through the barrier. Chiao and co-workers have more recently addressed the realization of superluminal speeds in a more systematic way. They used a periodic potential barrier to demonstrate experimentally that superluminal velocities can indeed be obtained, and showed that this result does not violate causality. In this article we intend to address the following questions: Can one trust the predictions of a nonrelativistic theory at all if superluminal effects are being investigated? How accurate are these predictions? Does the relativistic quantum theory predict superluminal speeds? Does a fully relativistic treatment of tunneling increase or reduce the tunneling probability? Does the existence of superluminal velocities imply the violation of Einstein’s causality when they are computed .