  We consider the production of a Standard-Model (SM) Higgs boson in gluon
fusion in hadronic collisions and compute the QCD corrections up to
next-to-next-to-leading order (NNLO) and match them to parton showers
(NNLO+PS). The complete dependence on the top-quark mass is taken into account
without making any approximations to the top quark loops mediating the coupling
between the gluons and the Higgs boson. To this end, we have included the
$gg\to H$ amplitudes up to three loops and the $pp\to H$+jet amplitudes up to
two loops in the full SM theory. This is the first fully differential
calculation of the top-quark mass effects up to NNLO in QCD, and we study their
impact on relevant observables for the LHC.
