  We consider various notions of vanishing mean oscillation on a (possibly
unbounded) domain $\Omega \subset \mathbb{R}^n$, and prove an analogue of
Sarason's theorem, giving sufficient conditions for the density of bounded
Lipschitz functions in the nonhomogeneous space $\rm{vmo}(\Omega)$. We also
study $\rm{cmo}(\Omega)$, the closure in $\rm{bmo}(\Omega)$ of the continuous
functions with compact support in $\Omega$. Using these approximation results,
we prove that there is a bounded extension from $\rm{vmo}(\Omega)$ and
$\rm{cmo}(\Omega)$ to the corresponding spaces on $\mathbb{R}^n$, if and only
if $\Omega$ is a locally uniform domain.
