Let k be a field of characteristic not 2, and let L be a Galois extension of k with group G. Let us denote by q_L its trace form. Then q_L is a G-form, and it is easy to see that L has a self-dual normal basis over k if and only if q_L is isomorphic to the unit G-form. The aim of this talk is to present some results and perspectives related to this topic. One of the tools we use is algebras with involution, and looking for analogs of Voevodsky's result (Milnor's conjecture) for these.