We consider Higgs boson production by gluon fusion in hadron collisions. We study the doubly-differential transverse-momentum (q T ) and rapidity (y) distribution of the Higgs boson in perturbative QCD. In the region of small q T (q T ≪ M H , M H being the mass of the Higgs boson), we include the effect of logarithmicallyenhanced contributions due to multiparton radiation to all perturbative orders. We use the impact parameter and double Mellin moments to implement and factorize the multiparton kinematics constraint of transverse-and longitudinal-momentum conservation. The logarithmic terms are then systematically resummed in exponential form. At small q T , we perform the all-order resummation of large logarithms up to next-to-next-to-leading logarithmic accuracy, while at large q T (q T ∼ M H ), we apply a matching procedure that recovers the fixed-order perturbation theory up to nextto-leading order. We present quantitative results for the differential cross section in q T and y at the LHC, and we comment on the comparison with the q T cross section integrated over y.