In comparative thanatology, most reports for nonhuman mammals concern mothers' behavioural responses to their dead offspring: most prominently, dead-infant carrying (sometimes of extended duration); but also inspection, proximity, maternal care such as grooming, protective behaviours and filial cannibalism. Documented across many primate species, these behaviours remain poorly understood in all. The literature is dominated by relatively brief qualitative descriptions of isolated anecdotal cases in apes and monkeys. We argue for quantitative coding in case reports, alongside analyses of longitudinal records of such events to allow objective evaluation of competing theories, and systematic comparisons within and across species and populations. Obtaining necessary datasets depends on raised awareness in researchers of the importance of recording occurrences and knowledge of pertinent data to collect. We review proposed explanatory hypotheses and outline data needed to test each empirically. To determine factors influencing infant-corpse carriage, we suggest analyses of deaths resulting in 'carry' versus 'no carry'. For individual cases, we highlight behavioural variables to code and the need for hormonal samples. We discuss mothers' stress and welfare in relation to infant death, continued transportation and premature removal of the corpse. Elucidating underlying proximate and ultimate causes is important for understanding phylogeny of maternal responses to infant death.This article is part of the theme issue 'Evolutionary thanatology: impacts of the dead on the living in humans and other animals'., (© 2018 The Author(s).)