We prove that hyperelliptic curves over Q of genus at least 2 have only finitely many rational points. Actually, we prove this for a more general class of high-genus curves over Q, including all solvable covers of P^1.
But wait, don’t we already know that, by Faltings? Of course we do. So the point of the paper is to show that you can get this finiteness in a different way, via the non-abelian Chabauty method pioneered by Kim. And I think it seems possible in principle to get Faltings for all curves over Q this way; though I don’t know how to do it.