Do optimal shapes exist?

The study of shape optimization problems is a very wide field, both classical, as the isoperimetric problem and the Newton problem of the best aerodynamical shape show, and modern, for all the recent results obtained in the last two, three decades. The interesting feature is that the competing objects are shapes, i.e. domains of RN, instead of functions, as it usually occurs in problems of the calculus of variations. This constraint often produces additional difficulties that lead to a lack of existence of a solution and to the introduction of suitable relaxed formulations of the problem. However, in some few cases an optimal solution exists, due to the special form of the cost functional and to the geometrical restrictions on the class of competing domains. The purpose of this talk is to make a short survey of this question.