On the Zariski Cancellation Problem

Given complex affine algebraic varieties $X$ and $Y$, the general Zariski Cancellation Problem asks whether the existence 

of an isomorphism $X\times\mathbb{C}^n\cong Y\times\mathbb{C}^n$ implies that $X\cong Y$.

Or, in other words, whether varieties with isomorphic cylinders should be isomorphic. This occurs to be true for affine 

curves (Abhyankar, Eakin, and Heinzer $'72$)  and false for affine surfaces (Danielewski $'89$).

The special Zariski Cancellation Problem asks the same question provided that $Y=\mathbb{C}^k$. In this case, the answer 

is "yes" in dimension $k=2$ (Miyanishi-Sugie $'80$ and Fujita $'79$), and unknown in higher dimensions, where the situation 

occurs to be quite mysterious (indeed, over a field of positive characteristic, there is a recent counter-example due to Neena Gupta $'14$).

The birational counterpart of the special Zariski Cancellation Problem asks whether stable rationality implies rationality.  The answer 

occurs to be negative; the first counter-example was constructed by Beauville, Colliot-Th\'el\`ene, Sansuc, and Swinnerton-Dyer $'85$. 

We will survey on the subject, both on some classical results and on a very recent development, reporting in particular on a joint 

work with Hubert Flenner and Shulim Kaliman.