  Horava gravity is a proposal for a UV completion of gravitation obtained by
endowing the space-time manifold with a preferred foliation in space-like
hypersurfaces. This allows for a power-counting renormalizable theory free of
ghosts, at the cost of breaking local Lorentz invariance and diffeomorphism
invariance down to foliation preserving transformations. In this updated
review, we report the main successes and challenges of the proposal, discussing
the main features of the projectable and non-projectable versions of Ho\v rava
gravity. We focus in three main aspects: (i) the UV regime, discussing the
renormalizability and renormalization group flow of the projectable theory, as
well as the obstacles towards similar results in the non-projectable case; (ii)
the low energy phenomenology of both models, including the PN regime, the most
updated constraints in the parameter space of the theory, the structure of
black holes at low energies, and the possibility of dark matter emerging from
gravitational dynamics in the projectable model; and (iii) the specific
phenomena induced by higher derivatives, such as the possibility of
regularizing singularities, the dynamical behavior of solutions to dispersive
equations, and the emission of Hawking radiation by universal horizons.
