  We show that for every $n\geq 2$ and any $\epsilon>0$ there exists a compact
hyperbolic $n$-manifold with a closed geodesic of length less than $\epsilon$.
When $\epsilon$ is sufficiently small these manifolds are non-arithmetic, and
they are obtained by a generalised inbreeding construction which was first
suggested by Agol for $n=4$. We also show that for $n\geq 3$ the volumes of
these manifolds grow at least as $1/\epsilon^{n-2}$ when $\epsilon\to 0$.
