A search for weakly interacting massive dark-matter particles produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and missing transverse momentum are considered. The analysis uses $$36.1 \; \mathrm {fb}^{-1}$$ of proton–proton collision data recorded by the ATLAS experiment at $$\sqrt{s}=13$$ TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are interpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour-neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross-section of 300 times the predicted rate for mediators with masses between 10 and $$50\; \hbox {GeV}$$ and assuming a dark-matter mass of $$1 \;\hbox {GeV}$$ and unitary coupling. Constraints on colour-charged scalar simplified models are also presented. Assuming a dark-matter particle mass of $$35\; \hbox {GeV}$$ , mediator particles with mass below $$1.1\; \hbox {TeV}$$ are excluded for couplings yielding a dark-matter relic density consistent with measurements.