  We report on the first fully differential calculation for double Higgs boson
production through gluon fusion in hadron collisions up to
next-to-next-to-leading order (NNLO) in QCD perturbation theory. The
calculation is performed in the heavy-top limit of the Standard Model, and in
the phenomenological results we focus on pp collisions at 14 TeV. We present
differential distributions through NNLO for various observables including the
transverse-momentum and rapidity distributions of the two Higgs bosons. NNLO
corrections are at the level of 10%-25% with respect to the next-to-leading
order (NLO) prediction with a residual scale uncertainty of 5%-15% and an
overall mild phase-space dependence. Only at NNLO the perturbative expansion
starts to converge yielding overlapping scale uncertainty bands between NNLO
and NLO in most of the phase-space. The calculation includes NLO predictions
for pp -> HH+jet+X. Corrections to the corresponding distributions exceed 50%
with a residual scale dependence of 20%-30%.
