The Lévy Walk is the process with continuous sample paths which arises from consecutive linear motions of i.i.d. lengths with i.i.d. directions. Assuming speed and motions in the domain of -stable attraction, we prove functional limit theorems and derive governing pseudo-differential equations for the law of the walker’s position. Both Lévy Walk and its limit process are continuous and ballistic in the case . In the case , the scaling limit of the process is -stable and hence discontinuous. This result is surprising, because the scaling exponent on the process level is seemingly unrelated to the scaling exponent of the second moment. For , the scaling limit is Brownian motion.