We show that for every $n\geq 2$ and any $��>0$ there exists a compact hyperbolic $n$-manifold with a closed geodesic of length less than $��$. When $��$ 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 $1/��^{n-2}$ when $��\to 0$.

12 pages; revised following referee's comments; to appear in Algebraic and Geometric Topology