Composite materials are increasingly important in making high-performance products. However, contemporary composites manufacturing processes still encounter significant challenges that range from inherent material stochasticity to manufacturing process variabilities. This paper proposes a novel smart Industrial Internet of Things framework, which is also referred to as an Artificial Intelligence of Things (AIoT) framework for composites manufacturing. This framework improves production performance through real-time process monitoring and AI-based forecasting. It comprises three main components: (i) an array of temperature, heat flux, dielectric, and flow sensors for data acquisition from production machines and products being made, (ii) an IoT-based platform for instantaneous sensor data integration and visualisation, and (iii) an AI-based model for production process forecasting. Via these components, the framework performs real-time production process monitoring, visualisation, and prediction of future process states. This paper also presents a proof-of-concept implementation of the framework and a real-world composites manufacturing case study that showcases its benefits.