The traditional production paradigm of large batch production does not offer flexibility towards satisfying the requirements of individual customers. A new generation of smart factories is expected to support new multi-variety and small-batch customized production modes. For that, Artificial Intelligence (AI) is enabling higher value-added manufacturing by accelerating the integration of manufacturing and information communication technologies, including computing, communication, and control. The characteristics of a customized smart factory are to include self-perception, operations optimization, dynamic reconfiguration, and intelligent decision-making. The AI technologies will allow manufacturing systems to perceive the environment, adapt to external needs, and extract the processed knowledge, including business models, such as intelligent production, networked collaboration, and extended service models. This paper focuses on the implementation of AI in customized manufacturing (CM). The architecture of an AI-driven customized smart factory is presented. Details of intelligent manufacturing devices, intelligent information interaction, and the construction of a flexible manufacturing line are showcased. The state-of-the-art AI technologies of potential use in CM, i.e., machine learning, multi-agent systems, Internet of Things, big data, and cloud-edge computing are surveyed. The AI-enabled technologies in a customized smart factory are validated with a case study of customized packaging. The experimental results have demonstrated that the AI-assisted CM offers the possibility of higher production flexibility and efficiency. Challenges and solutions related to AI in CM are also discussed.