We devote this first chapter to recalling the foundations for theoretical and numerical PDE-constrained optimization. First, we introduce the FreeFEM software and the finite element method for numerically solving the PDE constraint. We then present the theoretical framework for optimal control of PDEs and highlight the need for an adjoint variable to compute the derivative of the objective function. We then focus on the interior point method IpOpt and show with a simple example how it can be called from FreeFEM. Finally, we discuss different discretization strategies, distinguishing between the choices first optimize then discretize and first discretize then optimize, also showing how to use automatic differentiation.