This paper describes a search for events with one top-quark and large missing transverse momentum in the final state. Data collected during 2015 and 2016 by the ATLAS experiment from 13 TeV proton–proton collisions at the LHC corresponding to an integrated luminosity of 36.1 fb−1 are used. Two channels are considered, depending on the leptonic or the hadronic decays of the W boson from the top quark. The obtained results are interpreted in the context of simplified models for dark-matter production and for the single production of a vector-like T quark. In the absence of significant deviations from the Standard Model background expectation, 95% confidence-level upper limits on the corresponding production cross-sections are obtained and these limits are translated into constraints on the parameter space of the models considered.