This chapter examines the Dirichlet problem for second order elliptic operators in bounded domains. We explore the variational formulation of the problem and establish existence and uniqueness of solutions in Sobolev spaces. The chapter also addresses the regularity of solutions, showing that higher regularity of the domain, coefficients, and source terms leads to a corresponding improvement in the solution’s regularity. Finally, the discussion extends to a more general class of second-order elliptic operators, incorporating additional terms that allow a broader scope of applications. We will conclude the chapter by defining the Dirichlet to Neumann map and briefly discussing some application to an inverse problem.