The latest results in the search for rare and exotic Higgs boson decays are presented. The analyses are performed using proton-proton collision data collected with the ATLAS and CMS detectors at the LHC at a centre-of-mass energy $\sqrt{s} =$ 13 TeV. Several decay modes are reviewed, including very rare Standard Model decays as well as decays not predicted by this theory, such as lepton-flavour-violating decays and decays to non-Standard-Model particles. No evidence for new physics is found and the results are interpreted to set limits on the cross section times branching fraction of such decays.