Let $overline{M_{g, n}}$ be the moduli space of
stable curves of genus $g$ with $n$ marked points.
It is a classical problem in algebraic geometry
to determine which of these spaces are rational over $mathbb C$. 
In this talk, based on joint work with Matthieu Florence,
I will address the rationality problem for twisted forms
of $overline{M_{g, n}}$.  Twisted forms of $overline{M_{g, n}}$
are of interest because they shed light on the arithmetic
geometry of $overline{M_{g, n}}$,
and because they are coarse moduli spaces for natural moduli problems
in their own right. 

A classical result of Yu.~I.~Manin and P.~Swinnerton-Dyer
asserts that every form of $overline{M_{0, 5}}$ is rational.
(Recall that the $F$-forms $overline{M_{0, 5}}$ are precisely
the del Pezzo surfaces of degree $5$ defined over $F$.) 
Mathieu Florence and I recently showed that this theorem
continues to hold for forms of  $overline{M_{0, n}}$ if
$n geq 5$ is odd but fails if $n$ is even.  We also have 
similar results for forms of $overline{M_{g, n}}$, 
where $g leq 5$ (and $n$ is small). Our approach is based on reducing
the rationality problem for twisted forms of $overline{M_{g, n}}$
to certain special cases of the Noether problem. In the talk I plan
to survey some background material on $overline{M_{g, n}}$ and 
twisted forms, and explain how the Noether problem 
arises in this context.