  We consider Drell-Yan lepton pairs produced in hadronic collisions. We
present high-accuracy QCD predictions for the transverse-momentum ($q_T$)
distribution and fiducial cross sections in the small $q_T$ region. We resum to
all perturbative orders the logarithmically enhanced contributions up to the
next-to-next-to-next-to-next-to-leading logarithmic (N$^4$LL) accuracy and we
include the hard-virtual coefficient at the next-to-next-to-next-to-leading
order (N$^3$LO) (i.e. $\mathcal{O}(\alpha_S^3)$) with an approximation of the
N$^4$LO coefficients. The massive axial-vector and vector contributions up to
three loops have also been consistently included. The resummed partonic cross
section is convoluted with approximate N$^3$LO parton distribution functions.
We show numerical results at LHC energies of resummed $q_T$ distributions for
$Z/\gamma^*, W^\pm$ production and decay, including the $W^\pm$ and
$Z/\gamma^*$ ratio, estimating the corresponding uncertainties from missing
higher orders corrections and from incomplete or missing perturbative
information coefficients at N$^4$LL and N$^4$LO. Our resummed calculation has
been encoded in the public numerical program DYTurbo.
