Grothendieck envisaged a deep picture of a hypothetical category of pure and mixed motives of varieties which has partly been fullfilled so far. This entry gives only some pointers to approaches via dg-categories and their cousins, pretriangulated $A_\infty$-categories.
Maxim Kontsevich embeds the search for motives of varieties into the search for motives of the dg or $A_\infty$-enhancements of the derived categories of coherent sheaves on noncommutative varieties, what denotes in algebraic geometry working with a bit larger class of pretriangulated $A_\infty$-categories. There are also dg-approaches to Voevodsky’s derived category of mixed motives.
online video of Maxim Kontsevich’s talk at the conference for 61st birthday of Deligne, at IAS Princeton.
M.V. Bondarko, Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura, arXiv:math.AG/0601713
A. Beilinson, V. Vologodsky, A DG guide to Voevodsky’s motives, arXiv:math.KT/0604004
Marc Levine, Smooth motives, Motives and Algebraic Cycles. A Celebration in Honour of Spencer J. Bloch, Fields Institute Communications., 2009, arXiv.
Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435