Brownian motion is one of the most used stochastic models in applications to financial mathematics, communications, engineeering, physics and other areas. Many of the central results in the theory are obtained directly from its definition as a continuous process. As a mathematical object, Brownian motion also have some special and important properties that make it fundamental to understand related mathematical fields and state-of-the-art concepts. The purpose of this work is to review a relatively recent approach which allows to reobtain these results via a random walks approximation. The applications of this particular approach include the local time of Brownian motion and the Black-Scholes model in financial mathematics.