There is currently much controversy about which, if any, mental states chimpanzees and other nonhuman primates understand. In the current two studies we tested both chimpanzees' and human children's understanding of both knowledge-ignorance and false belief - in the same experimental paradigm involving competition with a conspecific. We found that whereas 6-year-old children understood both of these mental states, chimpanzees understood knowledge-ignorance but not false belief. After ruling out various alternative explanations of these and related findings, we conclude that in at least some situations chimpanzees know what others know. Possible explanations for their failure in the highly similar false belief task are discussed.