This paper presents measurements of tt¯ production in association with additional b-jets in pp collisions at the LHC at a centre-of-mass energy of 13 TeV. The data were recorded with the ATLAS detector and correspond to an integrated luminosity of 36.1 fb −1 . Fiducial cross-section measurements are performed in the dilepton and lepton-plus-jets tt¯ decay channels. Results are presented at particle level in the form of inclusive cross-sections of tt¯ final states with three and four b-jets as well as differential cross-sections as a function of global event properties and properties of b-jet pairs. The measured inclusive fiducial cross-sections generally exceed the tt¯ bb¯ predictions from various next-to-leading-order matrix element calculations matched to a parton shower but are compatible within the total uncertainties. The experimental uncertainties are smaller than the uncertainties in the predictions. Comparisons of state-of-the-art theoretical predictions with the differential measurements are shown and good agreement with data is found for most of them.[Figure not available: see fulltext.].