For a smooth variety X over a field k and a smooth k-group scheme G, Grothendieck and Serre predicted that every generically trivial G-torsor over X trivializes Zariski locally on X. I will explain a resolution of the Grothendieck--Serre problem, the main new case being when k is imperfect, in which pseudo-reductive and quasi-reductive groups play a central role. The argument is built on new purity and extension theorems for torsors valid for pseudo-finite, pseudo-proper, and pseudo-complete groups, and also rests on several other new results on algebraic groups in positive characteristic. The talk is based on joint work with Alexis Bouthier and Federico Scavia.