  A hyperelliptic curve over $\mathbb Q$ is called "locally soluble" if it has
a point over every completion of $\mathbb Q$. In this paper, we prove that a
positive proportion of hyperelliptic curves over $\mathbb Q$ of genus $g\geq 1$
are locally soluble but have no points over any odd degree extension of
$\mathbb Q$. We also obtain a number of related results. For example, we prove
that for any fixed odd integer $k > 0$, the proportion of locally soluble
hyperelliptic curves over $\mathbb Q$ of genus $g$ having no points over any
odd degree extension of $\mathbb Q$ of degree at most $k$ tends to 1 as $g$
tends to infinity. We also show that the failures of the Hasse principle in
these cases are explained by the Brauer-Manin obstruction. Our methods involve
a detailed study of the geometry of pencils of quadrics over a general field of
characteristic not equal to 2, together with suitable arguments from the
geometry of numbers.
