This paper, to appear in the proceedings of the October 24-27, 2001, Kinosaki Algebraic Geometry conference, is mostly a survey of Marc Levine's work on mixed Tate motives (Algebraic K-theory and algebraic topology (Lake Louise, 1991), 167--188, NATO Adv. Sci. Inst. Ser. C 407, Kluwer Acad. Publ., Dordrecht, 1993). It reports on a slight improvement obtained together with Annette Huber: an explicit formula for the weight filtration on the full subcategory generated by the Tate objects in Voevodsky's category of geometrical motives over a field, and some spectral sequences that can be deduced from it.