  In this paper we improve on existing methods to compute quadratic points on
modular curves and apply them to successfully find all the quadratic points on
all modular curves $X_0(N)$ of genus up to $8$, and genus up to $10$ with $N$
prime, for which they were previously unknown. The values of $N$ we consider
are contained in the set \[ \mathcal{L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100,
103, 107, 109, 113, 121, 127 \}.\] We obtain that all the non-cuspidal
quadratic points on $X_0(N)$ for $N\in \mathcal{L}$ are CM points, except for
one pair of Galois conjugate points on $X_0(103)$ defined over
$\mathbb{Q}(\sqrt{2885})$. We also compute the $j$-invariants of the elliptic
curves parametrised by these points, and for the CM points determine their
geometric endomorphism rings.
