We consider the possibility that the lightest supersymmetric particle is a heavy gluino. After discussing models in which this is the case, we demonstrate that the g̃-LSP could evade cosmological and other constraints by virtue of having a very small relic density. We then consider how neutral and charged hadrons containing a gluino will behave in a detector, demonstrating that there is generally substantial apparent missing momentum associated with a produced g̃-LSP. We next investigate limits on the g̃-LSP deriving from LEP, LEP2 and RunI Tevatron experimental searches for excess events in the jets plus missing momentum channel and for stable heavily-ionizing charged particles. The range of mg̃ that can be excluded depends upon the path length of the g̃ in the detector, the amount of energy it deposits in each hadronic collision, and the probability for the g̃ to fragment to a pseudo-stable charged hadron after a given hadronic collision. We explore how the range of excluded mg̃ depends upon these ingredients, concluding that for non-extreme cases the range 3 GeV <∼ mg̃ <∼ 130 − 150 GeV can be excluded at 95% CL based on currently available OPAL and CDF analyses. We find that RunII at the Tevatron can extend the excluded region (or discover the g̃) up to mg̃ ∼ 160 − 180 GeV. For completeness, we also analyze the case where the g̃ is the NLSP (as possible in gauge-mediated supersymmetry breaking) decaying via g̃ → g + gravitino. We find that the Tevatron RunI data excludes mg̃ ≤ 240 GeV. Finally, we discuss application of the procedures developed for the heavy g̃-LSP to searches for other stable strongly interacting particles, such as a stable heavy quark.