  We present a detailed study of Higgs boson production in association with a
single top quark at the LHC, at next-to-leading order accuracy in QCD. We
consider total and differential cross sections, at the parton level as well as
by matching short distance events to parton showers, for both t-channel and
s-channel production. We provide predictions relevant for the LHC at 13 TeV
together with a thorough evaluation of the residual uncertainties coming from
scale variation, parton distributions, strong coupling constant and heavy quark
masses. In addition, for t-channel production, we compare results as obtained
in the 4-flavour and 5-flavour schemes, pinning down the most relevant
differences between them. Finally, we study the sensitivity to a
non-standard-model relative phase between the Higgs couplings to the top quark
and to the weak bosons.
