Rationality and non-rationality of forms of the space $M_{g, n}$

In this talk we will show, following works by M. Florence and Z. Reichstein, that all forms of the space $M_{g, n}$ are rational for odd $n$, and this is not true for even $n$. Then we will discuss the stable rationality of forms of $M_{g, n}$ for small positive $g$ and some $n$.