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 Ho. rava 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.
