  Higgs boson production in association with a $Z$-boson at the LHC is
analysed, both in the Standard Model and in Simplified Model extensions for
Dark Matter. We focus on $H\rightarrow$invisibles searches and show that
loop-induced components for both the signal and background present
phenomenologically relevant contributions to the
$\mathcal{BR}(H\rightarrow\textit{inv})$ limits. In addition, the constraining
power of this channel to Simplified Models for Dark Matter with scalar and
pseudo-scalar mediators $\phi$ and $A$ is discussed and compared with
non-collider constraints. We find that with $100~fb^{-1}$ of LHC data, this
channel provides competitive constraints to the non-collider bounds, for most
of the parameter space we consider, bounding the universal Standard Model
fermion-mediator strength at $g_v < 1$ for moderate masses in the range of
${100~\text{GeV}<m_{\phi/A}<400}$ GeV.
