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Fast and heavy inorganic scintillators with suitable radiation tolerance are required to face the challenges presented at future hadron colliders of high energy and intensity. Up to 5 GGy and 5 × 1018 neq/cm2 of one-MeV-equivalent neutron fluence is expected by the forward calorimeter at the Future Hadron Circular Collider. This paper reports the results of an investigation of proton- and neutron-induced radiation damage in various fast and heavy inorganic scintillators, such as LYSO:Ce crystals, LuAG:Ce ceramics, and BaF2 crystals. The experiments were carried out at the Blue Room with 800 MeV proton fluence up to 3.0 × 1015 p/cm2 and at the East Port with one MeV equivalent neutron fluence up to 9.2 × 1015 neq/cm2, respectively, at the Los Alamos Neutron Science Center. Experiments were also carried out at the CERN PS-IRRAD proton facility with 24 GeV proton fluence up to 8.2 × 1015 p/cm2. Research and development will continue to develop LuAG:Ce ceramics and BaF2:Y crystals with improved optical quality, F/T ratio, and radiation hardness. [ABSTRACT FROM AUTHOR]