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.