brownian motion is a stochastic process named after botanist robert brown, who is credited with having discovered it in 1827 after observing the erratic behavior of pollen grains. it has since become one of the most important ideas in probability theory and has wide variety of applications everywhere from physics to finance. the talk will begin with a brief discussion of the necessary probability background, including the notions of expectation, martingales, and random walks. i will then define brownian motion and discuss some of its more interesting properties. if time permits, i will also discuss how brownian motion is used to define stochastic differential equations.