Mordell's conjecture on rational points of curves

Let $k$ be a number field and let $X$ be a smooth projective curve of genus $g\ge 2$ defined over $k$. Mordell's conjecture. The set of rational points $X(k)$ is finite. This is the finiteness statement now known as Faltings' theorem. It was proved by Faltings, so the conjecture is solved.