This chapter introduces Sobolev spaces, a framework for analyzing functions with weak derivatives and solving Partial Differential Equations in a more flexible way. These spaces allow reformulating classical problems, such as the Dirichlet problem, into a functional analysis setting that ensures well-posedness and unique solutions. In addition to foundational Sobolev spaces, the chapter touches on broader generalizations to provide a connection with existing mathematical literature. While not all material is immediately necessary for the book’s objectives, it offers valuable tools for understanding modern applications in analysis and beyond.