We review studies of single- and multiple-shot laser damage to plastics and theories of the damage mechanism. Single-shot damage is caused by absorbing defects, many of which can be eliminated by monomer purification. Multiple-shot damage is controlled by viscoelastic properties; plasticization improves damage resistance. Also, we review high-power laser studies of dye-impregnated plastics. Photobleaching is the most serious problem. We review progress in understanding the physics of that process. Dye-induced lowering of laser damage resistance and thermal bleaching are other important problems discussed. Plasticization to improve multiple-shot damage resistance may encourage thermal bleaching. Finally, we present a list of as yet untested and potentially damage-resistant commercially available plastics.