We consider the well-known Mittag−Leffler functions of one, two and three parameters, and establish some new connections between them using fractional calculus. In particular, we express the three-parameter Mittag−Leffler function as a fractional derivative of the two-parameter Mittag−Leffler function, which is in turn a fractional integral of the one-parameter Mittag−Leffler function. Hence, we derive an integral expression for the three-parameter one in terms of the one-parameter one. We discuss the importance and applications of all three Mittag−Leffler functions, with a view to potential applications of our results in making certain types of experimental data much easier to analyse.