A search for pair production of up-type vector-like quarks ( T ) with a significant branching ratio into a top quark and either a Standard Model Higgs boson or a Z boson is presented. The same analysis is also used to search for four-top-quark production in several new physics scenarios. The search is based on a dataset of pp collisions at √s = 13 TeV recorded in 2015 and 2016 with the ATLAS detector at the CERN Large Hadron Collider and corresponds to an integrated luminosity of 36.1 fb −1 . Data are analysed in the lepton+jets final state, characterised by an isolated electron or muon with high transverse momentum, large missing transverse momentum and multiple jets, as well as the jets + E T miss final state, characterised by multiple jets and large missing transverse momentum. The search exploits the high multiplicity of jets identified as originating from b -quarks, and the presence of boosted, hadronically decaying top quarks and Higgs bosons reconstructed as large-radius jets, characteristic of signal events. No significant excess above the Standard Model expectation is observed, and 95% CL upper limits are set on the production cross sections for the different signal processes considered. These cross-section limits are used to derive lower limits on the mass of a vector-like T quark under several branching ratio hypotheses assuming contributions from T → W b , Zt , Ht decays. The 95% CL observed lower limits on the T quark mass range between 0 . 99 TeV and 1 . 43 TeV for all possible values of the branching ratios into the three decay modes considered, significantly extending the reach beyond that of previous searches. Additionally, upper limits on anomalous four-top-quark production are set in the context of an effective field theory model, as well as in an universal extra dimensions model.