  Measurements of four-lepton differential and integrated fiducial
cross-sections in events with two same-flavour, opposite-charge electron or
muon pairs are presented. The data correspond to 139 fb$^{-1}$ of $\sqrt{s}=13$
TeV proton-proton collisions, collected by the ATLAS detector during Run 2 of
the Large Hadron Collider (2015-2018). The final state has contributions from a
number of interesting Standard Model processes that dominate in different
four-lepton invariant mass regions, including single $Z$ boson production,
Higgs boson production and on-shell $ZZ$ production, with a complex mix of
interference terms, and possible contributions from physics beyond the Standard
Model. The differential cross-sections include the four-lepton invariant mass
inclusively, in slices of other kinematic variables, and in different lepton
flavour categories. Also measured are dilepton invariant masses, transverse
momenta, and angular correlation variables, in four regions of four-lepton
invariant mass, each dominated by different processes. The measurements are
corrected for detector effects and are compared with state-of-the-art Standard
Model calculations, which are found to be consistent with the data. The
$Z\rightarrow 4\ell$ branching fraction is extracted, giving a value of
$\left(4.41 \pm 0.30\right) \times 10^{-6}$. Constraints on effective field
theory parameters and a model based on a spontaneously broken $B-L$ gauge
symmetry are also evaluated. Further reinterpretations can be performed with
the provided information.
