In this article we extend Voevodsky's nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky's original conjecture in the following cases: quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals, and Moishezon manifolds. , homological projective duality, Moishezon manifolds, noncommutative algebraic geometry.Sections 3-5 are a revised version of half of an old preprint [29]. The remaining work is a recent collaboration between the first and the last author.