Do non-human animals have minds? This scientific question, with important ethical implications, can be answered differently depending on the criteria used: in particular, whether we conceptualise animal minds through the lens of intelligence, or of feelings. This paper surveys the animal minds literature, and related philosophical issues. We review four core components of mind (consciousness, affective valence, agency, and intelligence), and their neurobiology. We survey current scientific evidence for language, tool use, problem-solving, theory of mind, self-awareness, and the behavioural and neuroscientific aspects of emotion. In overview: some animal species have fairly impressive intellectual abilities, but these are found only in a few species, often only in some species members, and only after substantial practice. Therefore, if mind is defined strictly by intelligence, then most non-human animals might be argued not to have minds. However, this paper argues that intelligence is a poor proxy for mind, and that we are better placed to debate the case for mind based on emotion. Viewed this way, there is evidence from a vast array of vertebrate species, where emotion is universal within those species, with none of the practice needed for intelligence. Jeremy Bentham was indeed correct when he pointed out that the critical issue is not merely the evidence, but the question of how we frame our thinking about non-human animals: are we to decide about animals based on their rationality, or on their capacity for feeling?