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 Z H production, the observed (expected) upper limit on the branching ratio of the Higgs boson to invisible particles is found to be 19% (19%) at the 95% confidence level. Exclusion limits are also set for simplified dark matter models and two-Higgs-doublet models with an additional pseudoscalar mediator.