Abstract Let g be a complex constant and z be complex variable, G(g) the Euler’s gamma function, and (g) k = G(g+k) G(g) for k 2 N[f0g the generalized Pochhammer symbol. The principal aim of this project is to investigate some properties of the Mittag-Leffler function defined by E g a;b (z) = ¥ å k=0 (g) G(ak +b) z k k k! ; where a; b; g 2 C and Â(a) > 0; Â(b) > 0; Â(g) > 0 and to study its relations with some other special functions of fractional calculus. The function E g a;b (z) is the generalization of the exponential function, the confluent hypergeometric function, the classical wright function, the Mittag-Leffler function of one and two parameters. For the function E g a;b (z), its various properties including Laplace transforms, Euler (Beta) transforms, Whittaker transforms, generalized hypergeometric series form, Mellin–Barnes integral representation with their some special cases are obtained. The usual differentiation and integration, fractional integration and differentiation of E (z) in Riemann-Liouville’s and Caputo’s senses are also obtained and relations with Confluent hypergeometric function, Meijer’s G-function, Fox H-function and Wright hypergeometric functions are verified. g a;b