The theoretical motivation for exotic stable massive particles (SMPs) and the results of SMP searches at non-collider facilities are reviewed. SMPs are defined such that they would be sufficiently long-lived so as to still exist in the cosmos either as Big Bang relics or secondary collision products, and sufficiently massive such that they are typically beyond the reach of any conceivable accelerator-based experiment. The discovery of SMPs would address a number of important questions in modern physics, such as the origin and composition of dark matter and the unification of the fundamental forces. This review outlines the scenarios predicting SMPs and the techniques used at non-collider experiments to look for SMPs in cosmic rays and bound in matter. The limits so far obtained on the fluxes and matter densities of SMPs which possess various detection-relevant properties such as electric and magnetic charge are given.