Electron and photon triggers covering transverse energies from 5 GeV to several TeV are essential for the ATLAS experiment to record signals for a wide variety of physics: from Standard Model processes to searches for new phenomena in both proton-proton and heavy-ion collisions. To cope with a fourfold increase of peak LHC luminosity from 2015 to 2018 (Run 2), to 2.1$\times$10$^{34}$cm$^{-2}$s$^{-1}$, and a similar increase in the number of interactions per beam-crossing to about 60, trigger algorithms and selections were optimised to control the rates while retaining a high efficiency for physics analyses. For proton-proton collisions, the single-electron trigger efficiency relative to a single-electron offline selection is at least 75% for an offline electron of 31 GeV, and rises to 96% at 60 GeV; the trigger efficiency of a 25 GeV leg of the primary diphoton trigger relative to a tight offline photon selection is more than 96% for an offline photon of 30 GeV. For heavy-ion collisions, the primary electron and photon trigger efficiencies relative to the corresponding standard offline selections are at least 84% and 95%, respectively, at 5 GeV above the corresponding trigger threshold.