  We review some recent results and conjectures saying that, roughly speaking,
periodic cyclic homology of a smooth non-commutative algebraic variety should
carry all the additional "motivic" structures possessed by the usual de Rham
cohomology of a smooth algebraic variety (specifically, an R-Hodge structure
for varieties over R, and a filtered Dieudonne module structure for varieties
over Z_p). To appear in Proc. ICM 2010.
