Mixed and pure motives, unsolved problems.
One hopes to develop the formalism of an abelian category of Mixed motives. This has not yet been done, but its derived category can be identified with Voevodsky motives (various versions); see also Cisinski-Déglise motives. Particular classes of mixed motives include Mixed Tate motives, Artin motives, 1-motives, and presumably Laumon 1-motives.
Mixed motives are intimately related to Motivic cohomology.
See also Pure motives, Chow motives, CM motives, Classical motives, Finite-dimensional motives, Grothendieck motives, Motives for absolute Hodge cycles, p-adic Chow motives, Sharp motives
See Motives for an overview over all different cats of motives (to appear).
Also t-motive, and ref to noncommutative motives. I think there is a DM category related to additive Chow groups (Park? Krishna??).
One could explain also the various versions of DM treated in TCMM.
A different story is Noncommutative motives
nLab page on D60 Motives