Experiments have observed a number of puzzling charmonium-like states in recent years. Of particular interest are states with ‘exotic flavour’, i.e. those with flavour quantum numbers which cannot arise from solely a quark-antiquark pair and so signal physics beyond a conventional meson. One proposed explanation for such states is that they are tetraquarks, compact states containing two quarks and two antiquarks. In Cheung et al [JHEP 1711 (2017) 033, arXiv:1709.01417] we developed methods to investigate the existence of tetraquarks using first-principles lattice QCD computations. The spectrum of states was calculated in a selection of exotic-flavour channels (isospin-1 hidden-charm and doubly-charmed channels), including a number where tetraquarks are expected in phenomenological models. There were no strong indications for any bound states or narrow resonances in the channels studied and so no sign of tetraquark candidates.