Parabolic partial differential equations are another important type of functional equations, including not only transportation but also diffusive components. Beyond the very frequent use in mathematical finance, they provide a powerful method for dealing with distributions in macroeconomics, in particular because of their relationship with stochastic differential equations via the Fokker-Planck-Kolmogorov equation. In this chapter, we solve and characterize linear scalar forward and backward equations, by using Fourier transform methods. A brief introduction to the dynamics of semi-linear equations can also be found in this chapter. At last, we show how parabolic partial differential equations can be used to model the dynamics of income distribution in the presence of both aggregate and pure social mobility.