The analysis of non linear optimal control problems and differential games via the solution of the Hamilton--Jacobi--Bellman or Isaacs equations was initiated by R. Bellman in the 60s. The method is based on Dynamic Programming and leads to the analysis and approximation of some non linear PDEs. The advantage of this approach is that it stands on solid mathematical grounds, the drawback is the difficulty to use it for large scale problems due to the curse of dimensionality. In the first part, I will present the basic ideas and show how the characterization of the value function can be derived via Dynamic Programming. I will also present very briefly the concept of weak solution in the viscosity sense, a notion which is very useful in this framework and has a great impact also in many other applications. The second part will be devoted to the numerical approximation of these problems, to the construction of the algorithms and to their analysis. I will present some numerical tests to show the main features (and limits) of this approach. No previous knowledge of control theory is required for this lectures.