In this chapter, we collect all required results from functional analysis, convex analysis, and measure theory. Moreover, we discuss the solution and regularity theory of the Poisson equation with either Dirichlet or Neumann boundary conditions (homogeneous and inhomogeneous), the Poisson equation with a right-hand side from a measure space, and the semilinear Poisson equation with a general monotone nonlinearity. For every result, we provide either a precise reference or proof if we cannot locate the results (or their proof) in the literature.