Brownian motion was originally observed by Robert Brown, who was examining pollen grains suspended in water under a microscope. This is an example of a random or stochastic process, which has found many applications: from describing the random erratic movement of molecules to the behavior of financial markets. In 1905, Einstein made a detailed study of Brownian motion in which he postulated certain properties (axioms) that should hold. In 1923, Norbert Wiener showed how to construct mathematically a random function W(t) giving the “position” of the molecules at time t which satisfied Einstein's axioms. We will talk about the fascinating history of Brownian motion and its applications.