  We review some applications of fractional calculus developed by the author
(partly in collaboration with others) to treat some basic problems in continuum
and statistical mechanics. The problems in continuum mechanics concern
mathematical modelling of viscoelastic bodies (Sect. 1), and unsteady motion of
a particle in a viscous fluid, i.e. the Basset problem (Sect. 2). In the former
analysis fractional calculus leads us to introduce intermediate models of
viscoelasticity which generalize the classical spring-dashpot models. The
latter analysis induces us to introduce a hydrodynamic model suitable to
revisit in Sect. 3 the classical theory of the Brownian motion, which is a
relevant topic in statistical mechanics. By the tools of fractional calculus we
explain the long tails in the velocity correlation and in the displacement
variance. In Sect. 4 we consider the fractional diffusion-wave equation, which
is obtained from the classical diffusion equation by replacing the first-order
time derivative by a fractional derivative of order $0< \beta <2$. Led by our
analysis we express the fundamental solutions (the Green functions) in terms of
two interrelated auxiliary functions in the similarity variable, which turn out
to be of Wright type (see Appendix), and to distinguish slow-diffusion
processes ($0 < \beta < 1$) from intermediate processes ($1 < \beta < 2$).
