Abstract: The fractional Brownian motion (fBm) is a Gaussian stochastic process indexed by a parameter $H in (0,1)$. The essential feature that makes fBm an appropriate model for many applications is the fact that it is an $H$-self-similar process whose increments are stationary and exhibit long-range dependence. If $H$ is not $1/2$, fBm is not a semimartingale and the Itô approach to the construction of a stochastic integral is not valid. In this talk, I will present the main properties of fBm, some of its applications and the basic theory of stochastic differential equations with respect to fractional Brownian motion.