Inclusive and differential fiducial cross sections of Higgs boson production in proton-proton collisions are measured in the H -> Z Z* -> 4l decay channel. The proton-proton collision data were produced at the Large Hadron Collider at a centre-of-mass energy of 13 TeV and recorded by the ATLAS detector in 2015 and 2016, corresponding to an integrated luminosity of 36.1 fb(-1). The inclusive fiducial cross section in the H -> Z Z* -> 4l decay channel is measured to be 3.62 +/- 0.50 (stat) (+0.25)(-0.20) (sys) fb, in agreement with the Standard Model prediction of 2.91 +/- 0.13 fb. The cross section is also extrapolated to the total phase space including all Standard Model Higgs boson decays. Several differential fiducial cross sections are measured for observables sensitive to the Higgs boson production and decay, including kinematic distributions of jets produced in association with the Higgs boson. Good agreement is found between data and Standard Model predictions. The results are used to put constraints on anomalous Higgs boson interactions with Standard Model particles, using the pseudo-observable extension to the kappa-framework.