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We give a review of UV renormalization of Ho\v{r}ava gravity (HG) modelsintroduced as a remedy against violation of unitarity in quantum gravitytheory. Projectable and non-projectable low-dimensional HG models and thespectra of their physical degrees of freedom are considered in the linearizedapproximation on flat spacetime background. The problem of regularity of theirpropagators depending on the choice of gauge fixing procedure is discussedalong with the role of this regularity in UV renormalization. With the choiceof a special class of quasi-relativistic gauge conditions perturbativerenormalizability of projectable HG models is proven in any spacetime dimensionand the status of renormalization in non-projectable models is brieflydiscussed. We show how the covariance of counterterms is provided within theclass 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 ofuniversal functional traces and obtain fixed points of its renormalizationgroup flow, some of them being good candidates for asymptotic freedom.

Ho\v{r}ava models as palladium of unitarity and renormalizability in quantum gravity