Physics case for an $e^{+}{e}^{-}$ collider at 500 GeV and above

Some highlights of the physics case for running an $e^{+}{e}^{-}$ collider at 500 GeV and above are discussed with a particular emphasis on the experimental access to the Higgs potential via di-Higgs and (at sufficiently high energy) triple Higgs production. The information obtainable from Higgs pair production at about 500 GeV is compared with the prospects for the HL-LHC and with the indirect information that can be obtained from a Higgs factory running at lower energies.