The top quark is the heaviest elementary particle and unique among the known quarks since it decays before forming hadronic bound states. Measurements involving top quarks in their final states provide precision tests of the Standard Model and are also sensitive to new physics at the high-energy frontier. The latest cross-section measurements of rare processes with top quarks ($tZq$, $t\gamma$$q$ and $t\bar{t}t\bar{t}$ production) as well as searches for flavour changing neutral currents are presented here using 13~TeV proton-proton collisions recorded by the ATLAS and CMS detectors at the Large Hadron Collider at CERN. The measured values are compared to the most accurate theoretical calculations. All these measurements probe the top quark couplings.}% Limits on the rate of flavour-changing neutral current processes in the production or decay of the top quark are also shown.