A landmark result of Birkar, Prokhorov, and Shramov shows that automorphism groups of Fano (or more generally rationally connected) varieties over C of a fixed dimension are uniformly Jordan.  This means in particular that there is some upper bound on the size of symmetric groups acting faithfully on rationally connected varieties of fixed dimension.  We give the first effective asymptotic bound on these symmetric group actions, as well as optimal bounds in all dimensions for special classes, such as Fano weighted complete intersections and toric varieties.  Finally, we show that klt Fano fourfolds with maximal symmetric actions are bounded, establishing a link between boundedness and large group actions. This talk is based on joint work with Lena Ji and Joaquín Moraga.