  By reformulating and extending results of Elkies, we prove some results on
$\mathbb Q$-curves over number fields of odd degree. We show that, over such
fields, the only prime isogeny degrees~$\ell$ which an elliptic curve without
CM may have are those degrees which are already possible over~$\mathbb Q$
itself (in particular, $\ell\le37$), and we show the existence of a bound on
the degrees of cyclic isogenies between $\mathbb Q$-curves depending only on
the degree of the field. We also prove that the only possible torsion groups of
$\mathbb Q$-curves over number fields of degree not divisible by a prime
$\ell\leq 7$ are the $15$ groups that appear as torsion groups of elliptic
curves over $\mathbb Q$. Complementing these theoretical results we give an
algorithm for establishing whether any given elliptic curve $E$ is a $\mathbb
Q$-curve, which involves working only over $\mathbb Q(j(E))$.
