  This is an expository article, which contributes to the Proceedings of the
conference "Groups of Automorphisms in Birational and Affine Geometry", held in
Trento in 2012. We propose that (rational) fibrations on the projective space
$\p^n$ by (birationally) Abelian hypersurfaces, for an arbitrary $n\ge 2$,
provide an obstruction to stable rationality of algebraic varieties. We discuss
the evidence for this proposition and derive some (almost straightforward)
corollaries from it.
