  The properties of Mittag-Leffler function is reviewed within the framework of
an umbral formalism. We take advantage from the formal equivalence with the
exponential function to define the relevant semigroup properties. We analyse
the relevant role in the solution of Schr\"odinger type and heat-type
fractional partial differential equations and explore the problem of
operatorial ordering finding appropriate rules when non-commuting operators are
involved. We discuss the coherent states associated with the fractional
Sch\"odinger equation, analyze the relevant Poisson type probability amplitude
and compare with analogous results already obtained in the literature.
