Measurements of the Higgs boson production times decay rates and differential cross-sections have recently been performed by the ATLAS experiment in several decay channels using up to 139 fb(-1) of proton-proton collision data at root s = 13TeV recorded at the Large Hadron Collider. This paper presents multiple interpretations of these Higgs boson measurements. Measurements of production-mode cross-sections, simplified template crosssections and fiducial differential cross-sections in different decay channels are reparameterised in terms of the impact of Standard Model effective field theory operators, and constraints are reported on the corresponding Wilson coefficients. Production and decay rate measurements are interpreted in UV-complete extensions of the Standard Model, namely the two-Higgsdoublet model (2HDM) near the alignment limit and the Minimal Supersymmetric Standard Model (MSSM) for various MSSM benchmark scenarios. The constraints on the 2HDM parameters (cos(beta - alpha), tan beta) and the MSSM parameters ( m(A), tan beta) are complementary to those obtained from direct searches for additional Higgs bosons.