H.S. Terrace and Peter Singer debate the evidence. One of Singer's core arguments:
In 2008, thirty years after David Premack and Guy Woodruff wrote a seminal paper entitled “Does the Chimpanzee Have a Theory of Mind?,” Josep Call and Michael Tomasello surveyed the extensive literature that now exists on this question. They conclude that “there is solid evidence from several different experimental paradigms that chimpanzees understand the goals and intentions of others, as well as the perception and knowledge of others.” Call and Tomasello do not find evidence that chimpanzees understand false beliefs, but it is clear that chimps have enough understanding of other minds to be able to perceive what others are thinking, and thus to use simple forms of language to communicate with them.
The discussion was sparked by a piece Singer wrote a few months ago on Project Nim, a film about Terrace's research. Recent coverage on animal minds here and here.