The theory of Brownian motion is an integral part of statistics and probability, and it also has some of the most diverse applications found in any topic in mathematics. With extensions into fields as vast and different as economics, physics, and management science, Brownian motion has become one of the most studied mathematical phenomena of the late twentieth and early twenty-first centuries. Today, Brownian motion is mostly understood as a type of mathematical process called a stochastic process. The word "stochastic" actually stems from the Greek word for "I guess," implying that stochastic processes tend to produce uncertain results, and Brownian motion is no exception to this, though with the right models, probabilities can be assigned to certain outcomes and we can begin to understand these complicated processes. This work reaches to attain this goal with regard to Brownian motion, and in addition it explores several applications found in the aforementioned fields and beyond.