The following four questions could be asked about each extremalproblem:1. What are the conditions for an extremum of theproblem (necessary, sufficient, necessary andsufficient)?2. How to describe the evolution of sulutions if theproblem is perturbed?3. Does a solution to the problem exist or not?4. How to find the solution numerically?The goal of these lecture notes is to discuss some importantgeneral approaches to these questions, to give sketches of proofsand to apply the general theory to some concrete problems.