We present charged-particle distributions sensitive to the underlying event, measured by the ATLAS detector in proton-proton collisions at a centre-of-mass energy of 13 TeV, in low-luminosity Large Hadron Collider fills corresponding to an integrated luminosity of 1.6 nb−1. The distributions were constructed using charged particles with absolute pseudorapidity less than 2.5 and with transverse momentum greater than 500 MeV, in events with at least one such charged particle with transverse momentum above 1 GeV. These distributions characterise the angular distribution of energy and particle flows with respect to the charged particle with highest transverse momentum, as a function of both that momentum and of charged-particle multiplicity. The results have been corrected for detector effects and are compared to the predictions of various Monte Carlo event generators, experimentally establishing the level of underlying-event activity at LHC Run 2 energies and providing inputs for the development of event generator modelling. The current models in use for UE modelling typically describe this data to 5% accuracy, compared with data uncertainties of less than 1%.