This thesis presents measurements of the Higgs boson inclusive and differential cross sections with 36.1 fb-1 of proton-proton collision data from the Large Hadron Collider at a center of mass energy of 13 TeV recorded with the ATLAS detector. The cross sections are measured in the fiducial phase space of the H ZZ* 4l (four lepton) decay channel as well as in the total Higgs phase space, combining data from the H ZZ* 4l and H [double gamma] (diphoton) decay channels. These two channels have high sensitivity to the Higgs signal and allow for full reconstruction of the Higgs boson kinematics. Unfolding techniques are used to correct for detector effects in the measurements, and comparisons are made to various state-of-the-art Standard Model theory predictions such as the predicted gluon-gluon fusion total cross section at N3LO and the Higgs differential transverse momentum spectrum at NNLO. These precision measurements are used to set limits on anomalous Higgs boson couplings beyond the Standard Model within the framework of pseudo-observables. In the H ZZ * 4l analysis, the inclusive fiducial Higgs boson cross section is measured to be 3.62 [plus/minus] 0.50 (stat) +0.25/-0.20 (sys) fb, in agreement with the Standard Model prediction of 2.91 [plus/minus] 0.13 fb. In the combined analysis, the inclusive total cross section is measured to be 57.0 +6.0/-5.9 (stat.) +4.0/-3.3 (syst.) pb, in agreement with the Standard Model prediction of of 55.6 [plus/minus] 2.5 pb. Measured differential cross sections are also presented for the Higgs transverse momentum, Higgs rapidity, number of associated jets, leading jet transverse momentum, and other kinematic variables.