We consider an emulsion whose droplets contain a trapped species (insoluble in the continuous phase) and study the emulsion＇s stability against coarsening via Lifshitz-Slyozov dynamics (Ostwald ripening). Extending an earlier treatment by Kabalnov et al. (Colloids Surf:, 1987, 24, 19-32), we derive a general condition on the mean initial droplet volume which ensures stability, even when arbitrary polydispersity is present in both size and composition of the initial droplets. We distinguish "nucleated" coarsening, which requires either fluctuations about the mean Geld equations or a tail in the initial droplet size distribution, from "spinodal" coarsening in which a typical droplet is locally unstable. A weaker condition for stability, previously suggested by Kabalnov et al., is sufficient only to prevent "spinodal" coarsening and is best viewed as a condition for metastability. The coarsening of unstable emulsions is considered and shown at long times to resemble that of ordinary emulsions (with no trapped species), but with a reduced value of the initial volume fraction of dispersed phase. We discuss the physical principles relevant to the stability of emulsions with trapped species, describing how these may be exploited to restabilize partially coarsened emulsions and to "shrink" previously formed emulsion droplets to form "miniemulsions".