This paper presents measurements of charged-particle distributions sensitive to the properties of the underlying event in events containing a Z boson decaying into a muon pair.The data were obtained using the ATLAS detector at the LHC in proton-proton collisions at a centre-of-mass energy of 13 TeV with an integrated luminosity of 3.2 fb−1. Distributions of the charged-particle multiplicity and of the charged-particle transverse momentum are measured in regions of the azimuth defined relative to the Z boson direction. The measured distributions are compared with the predictions of various Monte Carlo generators which implement different underlying-event models. The Monte Carlo model predictions qualitatively describe the data well, but with some significant discrepancies.