We investigate the monotonicity of the Mittag -Leffler function E with respect to the fractional parameter a. In particular we show that the mapping a  Ea (T(1 + a)r) decreases on (O, 1) for all  E R, that the mapping a  Ea(2) decreases on (0, as) for all a > 0 and some critical parameter as E (8,9) and that the mapping a - Ea(-a) decreaseson (0,1) for all   1 but not for all > 0. Analogous results are presented for the two-parameter Mittag-Leffier functions Ea. with B  a, with an emphasis on the extremal case B = a. Time permitting, we shall discuss several applications of these results to Abelian integral equations and subdiffusions. This is a joint work with Kenier Castillo (Coimbra) and Rui Ferreira (Porto).