Differential Higgs boson production cross sections are sensitive probes for physics beyond the Standard Model. New physics may contribute in the gluon-gluon fusion loop, the dominant Higgs boson production mechanism at the LHC, and manifest itself through deviations from the distributions predicted by the standard model. A variety of measurements are reported using the $\mathrm{H}\rightarrow{\rm \gamma}{\rm \gamma}$, $\mathrm{H}\rightarrow{\rm Z}{\rm Z^*}\rightarrow4\ell$, and their combinations together with boosted $\mathrm{H}\rightarrow{\rm b}{\rm \bar{b}}$ from ATLAS and CMS Collaborations. No significant deviations from the Standard Model expectations are observed. Precision on measurements is still largely statistically limited. Finally, projections of the differential cross section measurements for the High-Luminosity LHC are reported assuming different scenarios in the extrapolation of systematical uncertainties.