  Lorentz-symmetry and the notion of light cones play a central role in the
definition of horizons and the existence of black holes. Current observations
provide strong indications that astrophysical black holes do exist in Nature.
Here we explore what happens to the notion of a black hole in gravity theories
where local Lorentz symmetry is violated, and discuss the relevant
astrophysical implications. Einstein-aether theory and Horava gravity are used
as the theoretical background for addressing this question. We review earlier
results about static, spherically symmetric black holes, which demonstrate that
in Lorentz-violating theories there can be a new type of horizon and, hence, a
new notion of black hole. We also present both known and new results on slowly
rotating black holes in these theories, which provide insights on how generic
these new horizons are. Finally, we discuss the differences between black holes
in Lorentz-violating theories and in General Relativity, and assess to what
extent they can be probed with present and future observations.
