We consider extremum problems for some classical functionals of the Calculus of Variations (such as torsional rigidity, first Dirichlet eigenvalue, electrostatic capacity), subject to geometric constraints (such as prescribed volume, perimeter or mean width). For some of them the solution is well-known by classical methods, for some others it is conjectured but currently open. We shall try to overview some features and methods, which are related to the first order stationariety condition (involving overdetermined boundary value problems) and to second order properties (involving concavity inequalities for shapes).