Abstract: Many palatable animals, for example hoverflies, deter predators by mimicking well-defended insects such as wasps. However, for human observers, these flies often seem to be little better than caricatures of wasps—their visual appearance and behaviour are easily distinguishable from those which they are attempting to mimic. This imperfect mimicry baffles evolutionary biologists, because one might expect natural selection to do a more thorough job. Here we discuss two types of cognitive processes that might explain why distinguishable mimics could enjoy increased protection from predation. Speed–accuracy tradeoffs in predator decision making might give imperfect mimics sufficient time to escape, and predators under time constraint might avoid time-consuming discriminations between well-defended models and inaccurate edible mimics and instead adopt a “safety first” policy of avoiding insects with similar appearance. Categorisation of prey types by predators could mean that wholly dissimilar mimics may be protected, provided they share some common property with noxious prey. If predators use experience with multiple prey types to learn rules rather than just memorising the appearance of individual prey types, it follows that different individual predators should form different categories, each including separate types of novel prey. Experimental studies to test these hypotheses should be straightforward, because we can use the relatively simple signals (e.g., striped patterns) with which prey manipulate predator behaviour as tools for investigating cognitive processes that underlie decision making and object recognition in animals' daily lives.