  A search for heavy resonances decaying into a pair of $Z$ bosons leading to
$\ell^+\ell^-\ell'^+\ell'^-$ and $\ell^+\ell^-\nu\bar\nu$ final states, where
$\ell$ stands for either an electron or a muon, is presented. The search uses
proton-proton collision data at a centre-of-mass energy of 13 TeV collected
from 2015 to 2018 that corresponds to the full integrated luminosity of 139
fb$^{-1}$ recorded by the ATLAS detector during Run 2 of the Large Hadron
Collider. Different mass ranges spanning 200 GeV to 2000 GeV for the
hypothetical resonances are considered, depending on the final state and model.
In the absence of a significant observed excess, the results are interpreted as
upper limits on the production cross section of a spin-0 or spin-2 resonance.
The upper limits for the spin-0 resonance are translated to exclusion contours
in the context of Type-I and Type-II two-Higgs-doublet models, and the limits
for the spin-2 resonance are used to constrain the Randall--Sundrum model with
an extra dimension giving rise to spin-2 graviton excitations.
