It is widely agreed that all animals are entitled to some degree of welfare consideration, but that some are entitled to more consideration than others. However, the basis for singling out some animals for special consideration often seems to be mostly a matter of degree of similarity to, or association with, humans. A more reasonable criterion would involve the extent of suffering caused by given events. Two variables that seem likely to be very important in the extent of suffering are the capacity to anticipate and the capacity to recall. Everyday experience tells us that human suffering can be hugely amplified by either anticipation or recall of painful or distressing events. In the past, psychologists have tended to take the view that both these processes depend on the possession of language, and were therefore irrelevant to species other than humans. But comparative psychologists are increasingly making use of concepts from human cognition, including both memory and anticipation, to explain animals’ responses to both past and future events. These processes are invoked to explain the behaviour of a wide range of vertebrate species. Recent work on primate cognition indicates that more elaborate forms of representation may be possible in the great apes. Such evidence should be used as the basis for deciding whether to give special welfare consideration to certain species which have special cognitive capacities — or indeed enhanced welfare consideration to a wider range of species, if their cognitive capacities are found to be more sophisticated than is generally assumed.