An introduction to fractional Brownian motion and some of its applications is given. This introduction commences with some of the history and important properties of fractional Brownian motion. Some properties of a stochastic calculus for some fractional Brownian motions are described. Some stochastic differential equations in a finite dimensional space are given. Some results for linear and semilinear stochastic equations in a Hilbert space are introduced and some examples are provided. An absolute continuity of measures for fractional Brownian motion is presented. Some other applications of fractional Brownian motion are also described.