  Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in
1981 all possible cyclic isogenies of elliptic curves over $\mathbb{Q}$.
Although more than 40 years have passed, the determination of cyclic isogenies
of elliptic curves over a single other number field has hitherto not been
realised.
  In this paper we develop a procedure to assist in establishing such a
determination for a given quadratic field. Executing this procedure on all
quadratic fields $\mathbb{Q}(\sqrt{d})$ with $|d| < 10^4$ we obtain,
conditional on the Generalised Riemann Hypothesis, the determination of cyclic
isogenies of elliptic curves over $19$ quadratic fields, including
$\mathbb{Q}(\sqrt{213})$ and $\mathbb{Q}(\sqrt{-2289})$. To make this procedure
work, we determine all of the finitely many quadratic points on the modular
curves $X_0(125)$ and $X_0(169)$, which may be of independent interest.
