Parabolic partial differential equations are used to describe a wide variety of time-dependent phenomena, arising in a number of important physical problems. The aim of this talk is to present some semi-Lagrangian methods to approximate their solutions in some specific settings. First, we introduce a second order semi-Lagrangian method for advection-diffusion-reaction systems of equations on bounded domains, with Dirichlet boundary conditions. Afterwards, we present a first order semi-Lagrangian technique for approximating the solution to Hamilton-Jacobi-Bellman equations on bounded domains, with Neumann-type boundary conditions. Finally, we present a Lagrange-Galerkin approximation of the Fokker-Planck equation, and we show how to apply such a method to obtain a second-order accurate solution to Mean Field Games. Joint works with L. Bonaventura, E. Carlini, X. Dupuis, R. Ferretti and F.J. Silva.