Flexibility, unirationality and ellipticity after Mikhail Gromov

We define flexible affine varieties, discuss their properties and present many examples of flexible varieties. It is easy to see that any flexible variety is unirational. Bogomolov conjectured that up to stability any class of birational equivalence of unirational varieties contains a flexible variety. This conjecture is confirmed for several classes of birational equivalence.
We prove that a complete variety is unirational if and only if it admits a surjective morphism from an affine space. This is a joint result with Shulim Kaliman and Mikhail Zaidenberg. The proof is based on a study of elliptic varieties in the sense of Mikhail Gromov. Also we obtain a similar criterion of unirationality for affine cones.