Some celebrated conjectures of Beilinson, Grothendieck, Kimura, Tate, Voevodsky, Weil, and others, play a key central role in algebraic geometry. Notwithstanding the effort of several generations of mathematicians, the proof of (the majority of) these conjectures remains illusive. The aim of this course, prepared for a broad audience, is to give an overview of a recent noncommutative approach which has led to the proof of the aforementioned important conjectures in some new cases.