The monumental data collection campaign fielded by the particle physics experimenters at the LHC and future machines in the pipeline will bring the accuracy of measurements for the Higgs sector of the Standard Model to a much higher level than current theoretical predictions. One of the central variables in the study of Higgs boson production in proton collisions is its transverse momentum distribution. The calculation of this observable is complex due to the fact that the Higgs boson does not couple directly to the massless partons normally found in protons, but prefers to interact with intermediate high-mass virtual states such as the top quark and to a lesser extent the bottom quark. Furthermore, for the Higgs boson to have a transverse momentum, it must be produced together with another particle or a light hadron jet. In the talk we report the essential steps and results of the exact calculation of radiative corrections to the production of final states with a Higgs boson recoiling against a high-energy jet in proton-proton collisions at CERN's Large Hadron Collider. (based on https://arxiv.org/pdf/2206.10490)