Grothendieck’s standard conjectures, Voevodsky’s nilpotence conjecture, and Tate’s conjecture, play a key central role in algebraic geometry. Notwithstanding the effort of several generations of mathematicians, the proofs of these celebrated conjectures remain ellusive. The aim of this talk, prepared for a broad audience, is to give an overview of a recent topological/noncommutative approach which has led to the proof of the aforementioned important conjectures in several new cases.