  This survey covers some of the recent developments on noncommutative motives
and their applications. Among other topics, we compute the additive invariants
of relative cellular spaces and orbifolds; prove Kontsevich's semi-simplicity
conjecture; prove a far-reaching noncommutative generalization of the Weil
conjectures; prove Grothendieck's standard conjectures of type C+ and D,
Voevodsky's nilpotence conjecture, and Tate's conjecture, in several new cases;
embed the (cohomological) Brauer group into secondary K-theory; construct a
noncommutative motivic Gysin triangle; compute the localizing A1-homotopy
invariants of corner skew Laurent polynomial algebras and of noncommutative
projective schemes; relate Kontsevich's category of noncommutative mixed
motives to Morel-Voevodsky's stable A1-homotopy category, to Voevodsky's
triangulated category of mixed motives, and to Levine's triangulated category
of mixed motives; prove the Schur-finiteness conjecture for quadric fibrations
over low-dimensional bases; and finally extend Grothendieck's theory of periods
to the setting of dg categories.
