Most dog owners would insist that their dogs have a strong theory of mind, and I have read abstracts that support this idea - for instance, a dog who knows better than to snatch a piece of forbidden food from a plate on the floor while his or her master is looking becomes more likely to do so when the lights are turned off, which suggests that the dog understands that the owner can't see what he or she is doing in a dark room. I've also noticed that a dog will react differently when he or she is intentionally hurt by a mischievous child than he or she does when someone accidentally steps on his or her tail or paw. These phenomena suggest that dogs might understand that our perception is different from theirs, and that there is a difference between intentional harm and accidental harm.
Many of us would probably assume that higher primates are almost as skilled as we are in the use of the theory of mind, and again, I have seen research that seems to back this up. A chimp is placed in a room with plexiglass walls. The room faces a hallway. A hole in the wall allows the chimp to grab fruit from a tray in the hallway. A researcher walks into the hallway, picks up the tray, and throws it down the hall. The chimp goes ballistic. However, when the same process is repeated, but this time, the researcher pretends to accidentally trip over the tray, the chimp is less agitated, and instead of throwing a tantrum, he or she merely gestures towards the tray and waits to see if it will be returned to its original position. This suggests that chimps know the difference between accidents and intentions, which is not particularly surprising, considering how complex (and dangerous) chimpanzee society is.
Add to this anecdotal reports of whales and dolphins apparently showing gratitude when humans assist them (removing nets in which they've become tangled, for instance), and many similar accounts.
Do we have any idea which nonhuman animals possess a theory of mind, at least to some degree?