The Internet of Audio Things (IoAuT) is an emerging research field positioned at the intersection of the Internet of Things, sound and music computing, artificial intelligence, and human-computer interaction. The IoAuT refers to the networks of computing devices embedded in physical objects (Audio Things) dedicated to the production, reception, analysis and understanding of audio in distributed environments. Audio Things, such as nodes of wireless acoustic sensor networks, are connected by an infrastructure that enables multidirectional communication, both locally and remotely. In this paper, we first review the state of the art of this field, then we present a vision for the IoAuT and its motivations. In the proposed vision, the IoAuT enables the connection of digital and physical domains by means of appropriate information and communication technologies, fostering novel applications and services based on auditory information. The ecosystems associated with the IoAuT include interoperable devices and services that connect humans and machines to support human-human and human-machines interactions. We discuss challenges and implications of this field, which lead to future research directions on the topics of privacy, security, design of Audio Things, and methods for the analysis and representation of audio-related information.