Ho\v{r}ava Gravity at a Lifshitz Point: A Progress Report
  Ho\v{r}ava gravity at a Lifshitz point is a theory intended to quantize
gravity by using techniques of traditional quantum field theories. To avoid
Ostrogradsky's ghosts, a problem that has been plaguing quantization of general
relativity since the middle of 1970's, Ho\v{r}ava chose to break the Lorentz
invariance by a Lifshitz-type of anisotropic scaling between space and time at
the ultra-high energy, while recovering (approximately) the invariance at low
energies. With the stringent observational constraints and self-consistency, it
turns out that this is not an easy task, and various modifications have been
proposed, since the first incarnation of the theory in 2009. In this review, we
shall provide a progress report on the recent developments of Ho\v{r}ava
gravity. In particular, we first present four most-studied versions of
Ho\v{r}ava gravity, by focusing first on their self-consistency and then their
consistency with experiments, including the solar system tests and cosmological
observations. Then, we provide a general review on the recent developments of
the theory in three different but also related areas: (i) universal horizons,
black holes and their thermodynamics; (ii) non-relativistic gauge/gravity
duality; and (iii) quantization of the theory. The studies in these areas can
be generalized to other gravitational theories with broken Lorentz invariance.
