I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions could be asked over other fields, and they may even be more interesting in that situation. Let $M_g$ denote the moduli space of smooth projective curves of genus $g$. Is $M_g$ rational (i.e., birational to a projective space) for a general integer $g\geq 1$?

This is true when $g=1$, as $M_1$ is then just the moduli space of elliptic curves. I'm not sure how to show this for genus $2$, but note that as genus $2$ curves are hyperelliptic, the moduli space $M_2$ can be identified with $(\mathbf{P}^1)^6/(\mathrm{SL}_2\cdot\Sigma_6) = \mathbf{P}^6/\mathrm{SL}_2$, so it's at least unirational. (I think that this is even true when we work over a field of characteristic not $2$.) This suggests that the first step in proving rationality would be showing that $M_g$ is in general birationally equivalent to quotient of $\mathbf{P}^N$ by a reductive group, which implies that it is unirational. Rationality seems harder. The article https://arxiv.org/abs/0804.1509 shows that $M_3$ is also rational, and my thoughts about unirationality seem to be confirmed in the introduction. What about genus greater than $3$?