  This paper describes precision measurements of the transverse momentum
$p_\mathrm{T}^{\ell\ell}$ ($\ell=e,\mu$) and of the angular variable
$\phi^{*}_{\eta}$ distributions of Drell-Yan lepton pairs in a mass range of
66-116 GeV. The analysis uses data from 36.1 fb$^{-1}$ of proton-proton
collisions at a centre-of-mass energy of $\sqrt{s}=13$ TeV collected by the
ATLAS experiment at the LHC in 2015 and 2016. Measurements in electron-pair and
muon-pair final states are performed in the same fiducial volumes, corrected
for detector effects, and combined. Compared to previous measurements in
proton-proton collisions at $\sqrt{s}=$7 and 8 TeV, these new measurements
probe perturbative QCD at a higher centre-of-mass energy with a different
composition of initial states. They reach a precision of 0.2% for the
normalized spectra at low values of $p_\mathrm{T}^{\ell\ell}$. The data are
compared with different QCD predictions, where it is found that predictions
based on resummation approaches can describe the full spectrum within
uncertainties.
