Abstract: In this talk we will follow the paper "The initial-value problem for the Korteweg-de Vries equation" by Bona and Smith. Our aim is to focus on the basic concepts, including the statements of the problems, a brief history, the existence, and a priori estimates for the solutions. Then we will discuss about the stability problem. If time is allowed, we will include the comparison of solutions of the model invented by Benjamin and the KdV equation.