  Let k be a base field of positive characteristic. Making use of topological
periodic cyclic homology, we start by proving that the category of
noncommutative numerical motives over k is abelian semi-simple, as conjectured
by Kontsevich. Then, we establish a far-reaching noncommutative generalization
of the Weil conjectures, originally proved by Dwork and Grothendieck. In the
same vein, we establish a far-reaching noncommutative generalization of the
cohomological interpretations of the Hasse-Weil zeta function, originally
proven by Hesselholt. As a third main result, we prove that the numerical
Grothendieck group of every smooth proper dg category is a finitely generated
free abelian group, as claimed (without proof) by Kuznetsov. Then, we introduce
the noncommutative motivic Galois (super-)groups and, following an insight of
Kontsevich, relate them to their classical commutative counterparts. Finally,
we explain how the motivic measure induced by Berthelot's rigid cohomology can
be recovered from the theory of noncommutative motives.
