Ten years ago, Blanc and Furter proved that the group of birational transformations of the projective space (i.e. the Cremona group) of dimension $n\ge 2$ is not an (ind-)algebraic group. Since then, several new approaches have been developed to study maximal connected algebraic subgroups of Cremona groups, via birational geometry methods. In this talk, I will review recent progress in this area and present a joint work with E. Floris and S. Zimmermann, where we establish a link between the existence of non-rational stably rational varieties and the structure of maximal connected algebraic subgroups of Cremona groups.
