Arpanet, Internet, Internet of Services, Internet of Things, Internet of Skills. What next? We conjecture that in 15-20 years from now we will have the Internet of Neurons, a new Internet paradigm in which humans will be able to connect bi-directionally to the net using only their brain. The Internet of Neurons will provide new, tremendous opportunities thanks to constant access to unlimited information. It will empower all those outside of the technical industry, actually it will empower all human beings, to access and use technological products and services as everybody will be able to connect, even without possessing a laptop, a tablet or a smartphone. The Internet of Neurons will thus ultimately complete the currently still immature democratization of knowledge and technology. But it will also bring along several enormous challenges, especially concerning security (as well as privacy and trust). In this paper we speculate on the worldwide deployment of the Internet of Neurons by 2038 and brainstorm about its disruptive impact, discussing the main technological (and neurological) breakthroughs required to enable it, the new opportunities it provides and the security challenges it raises. We also elaborate on the novel system models, threat models and security properties that are required to reason about privacy, security and trust in the Internet of Neurons.