After decades of effort by some of our brightest human and non-human minds, there is still little consensus on whether or not non-human animals understand anything about the unobservable mental states of other animals or even what it would mean for a non-verbal animal to understand the concept of a 'mental state'. In the present paper, we confront four related and contentious questions head-on: (i) What exactly would it mean for a non-verbal organism to have an 'understanding' or a 'representation' of another animal's mental state? (ii) What should (and should not) count as compelling empirical evidence that a non-verbal cognitive agent has a system for understanding or forming representations about mental states in a functionally adaptive manner? (iii) Why have the kind of experimental protocols that are currently in vogue failed to produce compelling evidence that non-human animals possess anything even remotely resembling a theory of mind? (iv) What kind of experiments could, at least in principle, provide compelling evidence for such a system in a non-verbal organism?