Mazur's theorem for general number fields

Question

Mazur's theorem completely classifies the possibilities of $E_{tors}(\mathbb{Q})$ for an elliptic curve $E/\mathbb{Q}$, including the fact that only finitely many groups occur. What happens with $E_{tors}(k)$ for $E/k$ with $k$ a general number field? A bit of cursory searching suggests that at least the quadratic case is well-understood, but I don't know what the current state of the art is or how the proof of that case works. More specifically, my understanding of the proof of Mazur's theorem is that it involves proving that (1) $Y_1(N)$ parametrizes elliptic curves containing points of order exactly $N$; (2) these points are actually defined over $\mathbb{Q}$; (3) these rational points exist iff $X_1(N)$ has genus $0$. Do those ideas extend to more general cases?