The fractional Brownian motion is a centered self-similar Gaussian
process with stationary increments, which depends on a parameter H in
(0,1) called  the Hurst index. In  this talk we will describe some
basic properties of the fractional Brownian motion, and we  will
analyze different approaches  to construct a stochastic calculus with
respect to this process using path-wise techniques, Riemann sums  and
Malliavin calculus. We will  also present some  recent results on the
existence and uniqueness of solutions and numerical approximations
for stochastic differential equations driven by a fractional Brownian
motion.