  In Lovelock theories, gravity can travel faster or slower than light. The
causal structure is determined by the characteristic hypersurfaces. We
generalise a recent result of Izumi to prove that any Killing horizon is a
characteristic hypersurface for all gravitational degrees of freedom of a
Lovelock theory. Hence gravitational signals cannot escape from the region
inside such a horizon. We investigate the hyperbolicity of Lovelock theories by
determining the characteristic hypersurfaces for various backgrounds. First we
consider Ricci flat type N spacetimes. We show that characteristic
hypersurfaces are generically all non-null and that Lovelock theories are
hyperbolic in any such spacetime. Next we consider static, maximally symmetric
black hole solutions of Lovelock theories. Again, characteristic surfaces are
generically non-null. For some small black holes, hyperbolicity is violated
near the horizon. This implies that the stability of such black holes is not a
well-posed problem.
