This paper reports searches for heavy resonances decaying into $ZZ$ or $ZW$ using data from proton--proton collisions at a centre-of-mass energy of $\sqrt{s}=13$ TeV. The data, corresponding to an integrated luminosity of 36.1 fb$^{-1}$, were recorded with the ATLAS detector in 2015 and 2016 at the Large Hadron Collider. The searches are performed in final states in which one $Z$ boson decays into either a pair of light charged leptons (electrons and muons) or a pair of neutrinos, and the associated $W$ boson or the other $Z$ boson decays hadronically. No evidence of the production of heavy resonances is observed. Upper bounds on the production cross sections of heavy resonances times their decay branching ratios to $ZZ$ or $ZW$ are derived in the mass range 300--5000 GeV within the context of Standard Model extensions with additional Higgs bosons, a heavy vector triplet or warped extra dimensions. Production through gluon--gluon fusion, Drell--Yan or vector-boson fusion are considered, depending on the assumed model.
Selection acceptance times efficiency for ggF H -> Z Z -> llqq as a function of the Higgs boson mass, combining the HP and LP signal regions of the ZV -> llJ selection and the b-tagged and untagged regions of the ZV -> lljj selection.
Selection acceptance times efficiency for VBF H -> Z Z -> llqq as a function of the Higgs boson mass, combining the HP and LP signal regions of the ZV -> llJ selection and the b-tagged and untagged regions of the ZV -> lljj selection.
Selection acceptance times efficiency for ggF H -> Z Z -> vvqq as a function of the Higgs boson mass, combining the HP and LP signal regions.