Several versions of optimal control problems of parabolic partial differential equations are defined, and their optimality conditions are heuristically derived. This chapter concentrates on two optimal control problems. The first problem allows for solving a spatial version of the AK model of economic growth. The second problem is an optimal control problem of a Fokker-Planck-Kolmogorov equation associated to a diffusion process. As an application of this problem, we specify and solve an optimal social welfare problem, for an inequality averse social planner, when there is both aggregate and pure social mobility.