This talk was part of the Workshop on "New perspective on Shape and Topology Optimization" held at the ESI December 11 — 15, 2023.

Several results, mostly obtained in collaborations with Ana Cristina Barroso and José Matias, dealing with the optimal design problems, both in the framework of linear and nonlinear elasticity, will be discussed.
Among the several models presented I will focus also on the case in which the hyperelastic energy density has nonstandard growth. The case of thin structures will be also considered.