Let M(g, n) be the moduli space of stable curves of genus g with n marked points. A classical problem in algebraic geometry is to determine which of these spaces are rational. In this talk, based on joint work with Mathieu Florence, I will address the rationality problem for twisted forms of M(g, n). Twisted forms of M(g, n) are of interest because they shed light on the arithmetic geometry of M(g, n) and because they are coarse moduli spaces for natural moduli problems in their own right. A classical result of Enriques, Manin and Swinnerton-Dyer asserts that every form of M(0, 5) is rational. We showed that this theorem continues to hold for forms of if n >= 5 is odd but fails if n is even. We also have similar results for forms of M(g, n), where 1 <= g <= 5 (for small n).