A search for invisible decays of the Higgs boson as well as searches for dark matter candidates, produced together with a leptonically decaying $Z$ boson, are presented. The analysis is performed using proton-proton collisions at a centre-of-mass energy of 13 TeV, delivered by the LHC, corresponding to an integrated luminosity of 139 fb$^{-1}$ and recorded by the ATLAS experiment. Assuming Standard Model cross-sections for $ZH$ production, the upper limit on the branching ratio of the Higgs boson to invisible particles is found to be 19%. at the 95% confidence level. Exclusion limits are also set for simplified dark matter models and 2HDM$+a$ models.

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