This chapter considers an optimal control problem governed by the Poisson equation and subject to pointwise state constraints, i.e., inequality constraints for the state variable. Analysis and numerical analysis of such problems have a long history and have always been considered a challenge since the natural space for Lagrange multipliers related to the state constraints appears to be the space of regular Borel measures. We discuss optimality conditions and regularity issues. We especially provide an improved regularity result under smoothness assumptions on the state bounds. Moreover, we derive optimal order error estimates for the finite element discretization of this problem.