The Internet of Things (IoT) and Artificial Intelligence (AI) are revolutionizing the operation of systems, processes, infrastructure, and society, especially in the context of monitoring and prognosis. In these applications, it is crucial to adopt various types of sensors that can accurately assess the condition of systems and structures across extensive areas in a distributed manner and to effectively process multi-modal sensing data using AI algorithms. This enables the autonomous and intelligent operation of systems and structures in a safe and efficient way. To summarize recent progress and challenges, this paper presents a comprehensive review of the fundamentals of AI and IoT, their recent advancements, and their integration in monitoring and prognosis applications. Current issues, challenges, and future directions are discussed and summarized to highlight potential advancements in this rapidly evolving area. Overall, the combination of AI and IoT will significantly enhance the autonomy and intelligence in the monitoring and prognosis of systems and structures, leading to the era of the Artificial Intelligence of Things (AIoT).