The study of connected algebraic subgroups of the Cremona group is a classical way of deepening the understanding of the Cremona group. Via the Weil regularisation theorem and the Minimal Model Program, to such a group we associate a rational Mori fibre space on which it acts regularly. In this talk, we will discuss the notion of maximal connected algebraic subgroups of the Cremona group, and its relation with the geometry of the associated Mori fibre spaces. This is a work in collaboration with A. Fanelli and S. Zimmermann.