  We consider the recently proposed non-relativistic Ho\v{r}ava-Lifshitz
four-dimensional theory of gravity. We study a particular limit of the theory
which admits flat Minkowski vacuum and we discuss thoroughly the quadratic
fluctuations around it. We find that there are two propagating polarizations of
the metric. We then explicitly construct a spherically symmetric,
asymptotically flat, black hole solution that represents the analog of the
Schwarzschild solution of GR. We show that this theory has the same Newtonian
and post-Newtonian limits as GR and thus, it passes the classical tests. We
also consider homogeneous and isotropic cosmological solutions and we show that
although the equations are identical with GR cosmology, the couplings are
constrained by the observed primordial abundance of ${}^4{\rm He}$.
