  In this article, using the recent theory of noncommutative motives, we
compute the additive invariants of orbifolds (equipped with a sheaf of Azumaya
algebras) using solely "fixed-point data". As a consequence, we recover, in a
unified and conceptual way, the original results of Vistoli concerning
algebraic K-theory, of Baranovsky concerning cyclic homology, of the second
author with Polishchuk concerning Hochschild homology, and of Baranovsky-Petrov
and Caldararu-Arinkin (unpublished) concerning twisted Hochschild homology; in
the case of topological Hochschild homology and periodic topological cyclic
homology, the aforementioned computation is new in the literature. As an
application, we verify Grothendieck's standard conjectures of type C and D, as
well as Voevodsky's smash-nilpotence conjecture, in the case of
"low-dimensional" orbifolds. Finally, we establish a result of independent
interest concerning nilpotency in the Grothendieck ring of an orbifold.
