  In this article we prove that the Weinstein conjecture holds for contact
manifolds $(\Sigma,\xi)$ for which $\mathrm{Cont}_0(\Sigma,\xi)$ is
non-orderable in the sense of Eliashberg-Polterovich [EP00]. More precisely, we
establish a link between orderable and hypertight contact manifolds. In
addition, we prove for certain contact manifolds a conjecture by Sandon
[San13b] on the existence of translated points in the non-degenerate case.
