  We examine critically the theoretical underpinnings and phenomenological
implications of soft gluon (threshold) resummation of rapidity distributions at
a hadron collider, taking Drell-Yan production at the Tevatron and the LHC as a
reference test case. First, we show that in perturbative QCD soft gluon
resummation is necessary whenever the partonic (rather the hadronic)
center-of-mass energy is close enough to threshold, and we provide tools to
assess when resummation is relevant for a given process. Then, we compare
different prescriptions for handling the divergent nature of the series of
resummed perturbative corrections, specifically the minimal and Borel
prescriptions. We assess the intrinsic ambiguities of resummed results, both
due to the asymptotic nature of their perturbative expansion, and to the
treatment of subleading terms. Turning to phenomenology, we introduce a fast
and accurate method for the implementation of resummation with the minimal and
Borel prescriptions using an expansion on a basis of Chebyshev polynomials. We
then present results for W and Z production as well as both high- and low-mass
dilepton pairs at the LHC, and show that soft gluon resummation effects are
generally comparable in size to NNLO corrections, but sometimes affected by
substantial ambiguities.
