How non-human animals respond to dead or dying conspecifics, and what their responses may reveal about the extent to which a concept of “death" is present outside the hominin lineage, are questions currently generating intense scientific interest. Answers to these questions may help elucidate the evolutionary origins of a range of death-related psychological states and behaviours in our species, such as grief, mourning, understanding of mortality and mortuary practices. In addition, they raise welfare considerations in the case of captive animals, regarding the management of situations involving death or dying. I will describe a number of case studies from our closest evolutionary relatives, chimpanzees, together with a review of examples that have accumulated to date across a broad spectrum of mammalian species (including other primates, canids, ungulates, elephants, and cetaceans). Documented responses range from attending next to the deceased conspecific and vocal or behavioural indicators of distress, through focused curiosity directed at the body and active caretaking such as grooming, licking or carrying, to sexual behaviour and aggression. Drawing on this emerging body of evidence, I will explore what cues might elicit these responses, which factors might determine what combination of responses is exhibited, and what these may reveal about animals' underlying psychological states. In doing so, I will present an emerging framework for interpreting such behavioural observations, and the care we need to take to neither overinterpret nor too easily dismiss them as being relevant to humans' understanding of death.