  This is a survey of recent examples of varieties that are not stably
rational. We review the specialization method based on properties of the Chow
group of zero-cycles used in these examples and explain the point of view of
unramified cohomology for the construction of nontrivial stable invariants of
the special fiber. In particular, we find an explicit formula for the Brauer
group of fourfolds fibered in quadrics of dimension two over a rational
surface.
