Searches are presented for rare and lepton-flavour violating decays of the Higgs boson. The searches are performed using up to 139 fb$^{-1}$ of $\sqrt{s}=13$ TeV proton--proton collision data collected with the ATLAS detector at the Large Hadron Collider. The final states under study are: (i) a photon and a meson or a $Z$ boson, (ii) a charm-anticharm quark pair, (iii) a $\tau$ lepton and an electron or a muon, and (iv) a dimuon pair. No significant excess with respect to the expected background is observed in the data for any of these final states, and upper limits are set on the production cross section times decay branching ratio for each of them.