We and our colleagues have been doing studies of great ape gestural communication for more than 30 years. Here we attempt to spell out what we have learned. Some aspects of the process have been reliably established by multiple researchers, for example, its intentional structure and its sensitivity to the attentional state of the recipient. Other aspects are more controversial. We argue here that it is a mistake to assimilate great ape gestures to the species-typical displays of other mammals by claiming that they are fixed action patterns, as there are many differences, including the use of attention-getters. It is also a mistake, we argue, to assimilate great ape gestures to human gestures by claiming that they are used referentially and declaratively in a human-like manner, as apes' "pointing" gesture has many limitations and they do not gesture iconically. Great ape gestures constitute a unique form of primate communication with their own unique qualities.