Abstract: Let K be a number field and let G be a finite group. In the locally free class group Cl(O_K[G]) we can consider the subset of classes realized as Galois modules by the ring of integers of tamely ramified
G-extensions. The study of the structure of this subset goes under the name of realizable classes problem. In this talk, after a general presentation of the problem, I will give an overview of the known results and I shall point out the main recent open questions on this topic. No prior knowledge about locally free class groups and realizable classes will be assumed.