Fractional calculus has become quite popular in the scientific community due to its versatility to describe constitutive laws of materials and devices (viscoelastic materials, capacitors, electrochemical reactions and so on). After a brief presentation of the fractional operators the seminar will focus on two applications that are critical in the theory of stochastic processes and probabilities. -The first one is the representation of Power Spectral Density (PSD) and Correlation Function (CF) by the complex fractional spectral moments. Such quantities are generalizations of the classical spectral moments concept. It will be shown that the fractional spectral moments are strictly related to the Riesz fractional integrals. Further, they can be used for reconstructing both PSD and CF. -The second one is the representation of the probability density function and characteristic function by (complex) fractional moments. It will be shown that the fractional moments yield the probability density function and the corresponding characteristic function in the entire domain of interest; this property holds also for ?-stable random variables. Applications of the previous concepts to stochastic differential equations encountered in various engineering will also be discussed.