We review the theoretical motivations and experimental status of searches for stable massive particles (SMPs) which could be sufficiently long-lived as to be directly detected at collider experiments. The discovery of such particles would address a number of important questions in modern physics including the origin and composition of dark matter in the universe and the unification of the fundamental forces. This review describes the techniques used in SMP-searches at collider experiments and the limits so far obtained on the production of SMPs which possess various colour, electric and magnetic charge quantum numbers. We also describe theoretical scenarios which predict SMPs along with the phenomenology needed to model their production at colliders and interactions with matter. In addition, the interplay between collider searches and open questions in cosmology such as dark matter composition is addressed.