  We consider the transverse-momentum ($p_T$) distribution of $Z\gamma$ pairs
produced in hadronic collisions. Logarithmically enhanced contributions at
small $p_T$ are resummed to all orders in QCD perturbation theory and combined
with the fixed-order prediction. We achieve the most advanced prediction for
the $Z\gamma$ $p_T$ spectrum by matching next-to-next-to-next-to-leading
logarithmic (N$^3$LL) resummation to the integrated cross section at
next-to-next-to-leading order (NNLO). By considering $\ell^+\ell^-\gamma$
production at the fully differential level, including spin correlations,
interferences and off-shell effects, arbitrary cuts can be applied to the
leptons and the photon. We present results at the LHC in presence of fiducial
cuts and find agreement with the $13$\,TeV ATLAS data at the few-percent level.
