In this article, we introduce Mittag-Leffler Lévy process and provide two alternative representations of this process. First, in terms of Laplace transform of the marginal densities and next as a subordinated stochastic process. Both these representations are useful in analyzing the properties of the process. Since integer order moments for this process are not finite, we obtain fractional order moments. Its density function and corresponding Lévy measure density is also obtained. Further, heavy tailed behavior of densities and stochastic self-similarity of the process is demonstrated. Our results generalize and complement the results available on Mittag-Leffler distribution in several directions.