  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 $2\to 2$ 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.
