Theories beyond the Standard Model can predict particles with lifetimes much longer than those of known unstable particles. When produced at high-energy collisions, e.g. at the CERN LHC, these long-lived particles can decay far from the primary interaction vertex. Recently, a number of dedicated experiments have been proposed to search for such objects at the LHC: MoEDAL-MAPP, MoEDAL-MALL, FASER, CODEX-b, MATHUSLA, AL3X and ANUBIS. The SHiP beam-dump experiment, on the other hand, is planned to operate with the SPS beam to extend the discovery reach of such particles. The already operating MoEDAL detector, its MAPP extension and the planned milliQan experiment are sensitive to particles with exotic electromagnetic charges producing anomalous ionisation. The planned detector design and expected physics sensitivity of these experiments are presented here with emphasis on scenarios explaining the nature of dark matter and the origin of neutrino masses.