  In this survey, written for the proceedings of the conference K-theory in
algebra, analysis and topology, Buenos-Aires, Argentina (satellite event of the
ICM 2018), we give a rigorous overview of the noncommutative counterparts of
some celebrated conjectures of Grothendieck, Voevodsky, Beilinson, Weil, Tate,
Parshin, Kimura, Schur, and others.
