In the 1980s Beilinson conjectured the existence of motivic cohomology, together with certain precise expected relations to algebraic K-theory, étale cohomology, algebraic cycles, and the category of mixed motives. Apart from the still conjectural category of mixed motives (and its consequences such as the Beilinson--Soulé vanishing conjecture), Beilinson's vision was essentially fulfilled by Voevodsky in the case of smooth algebraic varieties. I will overview recent work which yields a theory of motivic cohomology for arbitrary schemes. Moreover, by resolving several conjectures of Voevodsky, its A^1-localisation is shown to arise from a candidate derived category of mixed motives. This work is joint with Bachmann--Elmanto, or with Elmanto, or due to Bouis.