In this chapter we show the existence of solutions to the basic boundary value problems for the Laplace equation using a classical approach based on Potential Theory. Specifically, we consider the Dirichlet problem, the Neumann problem, the Robin problem, the transmission problem, and a mixed problem. To do so, we carry out an analysis of the boundary integral operators associated with the single and double layer potentials in a Schauder space setting. Our presentation of the topic stems from that of Folland (Introduction to partial differential equations. Princeton University Press, Princeton, NJ, second edition, 1995, Chap. 3).