  We have revisited the Brownian motion on the basis of the fractional Langevin
equation which turns out to be a particular case of the generalized Langevin
equation introduced by Kubo on 1966. The importance of our approach is to model
the Brownian motion more realistically than the usual one based on the
classical Langevin equation, in that it takes into account also the retarding
effects due to hydrodynamic backflow, i.e. the added mass and the Basset memory
drag. On the basis of the two fluctuation-dissipation theorems and of the
techniques of the Fractional Calculus we have provided the analytical
expressions of the correlation functions (both for the random force and the
particle velocity) and of the mean squared particle displacement. The random
force has been shown to be represented by a superposition of the usual white
noise with a "fractional" noise. The velocity correlation function is no longer
expressed by a simple exponential but exhibits a slower decay, proportional to
$t^{-3/2}$ as $t \to \infty$, which indeed is more realistic. Finally, the mean
squared displacement has been shown to maintain, for sufficiently long times,
the linear behaviour which is typical of normal diffusion, with the same
diffusion coefficient of the classical case. However, the Basset history force
induces a retarding effect in the establishing of the linear behaviour, which
in some cases could appear as a manifestation of anomalous diffusion to be
correctly interpreted in experimental measurements.
