We investigate the monotonicity of the Mittag-Leffler functionEαwith respect to thefractional parameterα. In particular, we show that the mappingα 7→Eα(Γ(1 +α)x)decreases on(0,1) for allx∈R∗, that the mappingα 7→Eα(xα)decreases on(0,α∗) forallx >0and some critical parameterα∗∈(8,9)and that the mappingα 7→Eα(−xα)decreases on(0,1) for allx≥1but not for allx >0. Analogous results are presentedfor the two parameter Mittag-Leffler functionsEα,βwithβ≥α, with an emphasis onthe extremal caseβ=α. Time permitting, we shall discuss several applications of theseresults to Abelian integral equations and subdiffusions. This is a joint work with KenierCastillo (Coimbra) and Rui Ferreira (Porto).