  We give a review of UV renormalization of Ho\v{r}ava gravity (HG) models
introduced as a remedy against violation of unitarity in quantum gravity
theory. Projectable and non-projectable low-dimensional HG models and the
spectra of their physical degrees of freedom are considered in the linearized
approximation on flat spacetime background. The problem of regularity of their
propagators depending on the choice of gauge fixing procedure is discussed
along with the role of this regularity in UV renormalization. With the choice
of a special class of quasi-relativistic gauge conditions perturbative
renormalizability of projectable HG models is proven in any spacetime dimension
and the status of renormalization in non-projectable models is briefly
discussed. We show how the covariance of counterterms is provided within the
class of background covariant gauge conditions and show asymptotic freedom of
$(2+1)$-dimensional HG. We also present the calculation of beta-functions in
$(3+1)$-dimensional theory by the generalized Schwinger-DeWitt technique of
universal functional traces and obtain fixed points of its renormalization
group flow, some of them being good candidates for asymptotic freedom.
