A search for the production of three massive vector bosons in proton–proton collisions is performed using data at s=13 TeV recorded with the ATLAS detector at the Large Hadron Collider in the years 2015–2017, corresponding to an integrated luminosity of 79.8 fb−1. Events with two same-sign leptons ℓ (electrons or muons) and at least two reconstructed jets are selected to search for WWW→ℓνℓνqq. Events with three leptons without any same-flavour opposite-sign lepton pairs are used to search for WWW→ℓνℓνℓν, while events with three leptons and at least one same-flavour opposite-sign lepton pair and one or more reconstructed jets are used to search for WWZ→ℓνqqℓℓ. Finally, events with four leptons are analysed to search for WWZ→ℓνℓνℓℓ and WZZ→qqℓℓℓℓ. Evidence for the joint production of three massive vector bosons is observed with a significance of 4.1 standard deviations, where the expectation is 3.1 standard deviations.