  This survey paper reports on work of Birkar, who confirmed a long-standing
conjecture of Alexeev and Borisov-Borisov: Fano varieties with mild
singularities form a bounded family once their dimension is fixed. Following
Prokhorov-Shramov, we explain how this boundedness result implies that
birational automorphism groups of projective spaces satisfy the Jordan
property, answering a question of Serre in the positive.
