We analyse the associated production of Higgs and $Z$ boson via heavy-quark loops at the LHC in the Standard Model and beyond. We first review the main features of the Born $2to 2$ production, and in particular discuss the high-energy behaviour, angular distributions and $Z$ boson polarisation. We then consider the effects of extra QCD radiation as described by the $2 to 3$ loop matrix elements, and find that they dominate at high Higgs transverse momentum. We show how merged samples of 0-- and 1--jet multiplicities, matched to a parton shower can provide a reliable description of differential distributions in $ZH$ production. In addition to the Standard Model study, results in a generic two-Higgs-doublet-model are obtained and presented for a set of representative and experimentally viable benchmarks for $Zh^0$, $ZH^0$ and $ZA^0$ production. We observe that various interesting features appear either due to the resonant enhancement of the cross-section or to interference patterns between resonant and non-resonant contributions.
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