For acquired behaviour to count as cultural, two conditions must be met: it must propagate in a social group, and it must remain stable in the process of propagation. It is commonly claimed that imitation is the mechanism that explains both the spread of animal culture and its persistence. We review the literature on transmission chain studies in chimpanzees and other animals. We use a formal model to argue, that imitation, which may well play a major role in the propagation of animal culture, cannot be considered faithful enough to explain its stability. We show that adding to the capacity for imitation a relatively strong conformist tendency of the kind suggested by Boyd and Richerson is relevant but unlikely to be sufficient. We consider the contribution that other psychological or ecological factors might make to the persistence of animal culture observed in the wild.