Severi proved that the moduli space of curves $\mathcal{M}_g$ is unirational when $g$ is at most $10$. This has now been made rigorous. Severi further conjectured that the moduli space is unirational for all values of $g$, but this was famously disproved by Eisenbud, Harris, and Mumford. They prove that $\overline{\mathcal{M}_g}$ is of general type when $g \geq 24$. Farkas has shown that it is of general type when $g = 22$. It is known that when $g \leq 14$ the moduli space is unirational, but I believe that for remaining values of $g$, this problem is still open.

Severi proved that the moduli space of curves $M_g$ is unirational when $g$ is at most $10$. This has now been made rigorous. Severi further conjectured that the moduli space is unirational for all values of $g$, but this was famously disproved by Eisenbud, Harris, and Mumford. They prove that $\overline{M}_g$ is of general type when $g \geq 24$. Farkas has shown that it is of general type when $g = 22$. It is known that when $g \leq 14$ the moduli space is unirational, but I believe that for remaining values of $g$, this problem is still open.