In 1980, Robert Seyfarth, Dorothy Cheney and Peter Marler published a landmark paper in Science claiming language-like semantic communication in the alarm calls of vervet monkeys. This article and the career research program it spawned for its authors catalyzed countless other studies searching for semantics, and then also syntax and other rarefied properties of language, in the communication systems of non-human primates and other animals. It also helped bolster a parallel tradition of teaching symbolism and syntax in artificial language systems to great apes. Although the search for language rudiments in the communications of primates long predates the vervet alarm call story, it is difficult to overstate the impact of the vervet research, for it fueled field and laboratory research programs for several generations of primatologists and kept busy an equal number of philosophers, linguists, and cognitive scientists debating possible implications for the origins and evolution of language and other vaunted elements of the human condition. Now 40-years on, the original vervet alarm call findings have been revised and claims of semanticity recanted; while other evidence for semantics and syntax in the natural communications of non-humans is sparse and weak. Ultimately, we are forced to conclude that there are simply few substantive precedents in the natural communications of animals for the high-level informational and representational properties of language, nor its complex syntax. This conclusion does not mean primates cannot be taught some version of these elements of language in artificial language systems – in fact, they can. Nor does it mean there is no continuity between the natural communications of animals and humans that could inform the evolution of language – in fact, there is such continuity. It just does not lie in the specialized semantic and syntactic properties of language. In reviewing these matters, I consider why it is that primates do not evince high-level properties of language in their natural communications but why we so readily accepted that they did or should; and what lessons we might draw from that experience. In the process, I also consider why accounts of human-like characteristics in animals can be so irresistibly appealing.