We study limit laws for simple random walks on supercritical long range percolation clusters on Z^d. For the long range percolation model, the probability that two vertices x, y are connected is asymptotically |x-y|^{-s}. For d < s < d+1 we prove that the scaling limit of simple random walk on the infinite component is \alpha-stable Levy motion with alpha = s-d establishing a conjecture of Berger...