We will recall how the classification of the subgroups of Cremona up to conjugation relies on the study of groups of symmetries of rational varieties and equivariant birational equivalence. Two questions currently have a particular interest for researchers. A subgroup G of Cremona is linearizableif it is conjugated to a group of Automorphisms of the projective space. It is solid if, after regularization and G-MMP, it can only act on a Fano variety with invariant class group of rank one.

We will give the complete answer to both questions in dimension 2. The answer for linearizability is a joint work with A.Sarikyan and E.Yasinsky.