This talk was part of the Workshop on "New perspective on Shape and Topology Optimization" held at the ESI December 11 — 15, 2023. It is well-known that one of the main problems in the stability of suspension bridges is the appearance of unexpected and destructive torsional oscillations. Based on different mathematical models for bridges we propose some optimization problems aiming to improve the stability of the structure by modifying some of its parameters, such as the distribution of material in the deck, the shape of its cross-sections, the location of piers. Some characterization results for the extremizers are stated together with several open problems. This study was carried out within the PRIN project 201758MTR2: “Direct and inverse problems for partial differential equations: theoretical aspects and applications” (Italy)