By interacting, synchronizing, and cooperating with its physical counterpart in real time, digital twin is promised to promote an intelligent, predictive, and optimized modern city. Via interconnecting massive physical entities and their virtual twins with inter-twin and intra-twin communications, the Internet of digital twins (IoDT) enables free data exchange, dynamic mission cooperation, and efficient information aggregation for composite insights across vast physical/virtual entities. However, as IoDT incorporates various cutting-edge technologies to spawn the new ecology, severe known/unknown security flaws and privacy invasions of IoDT hinders its wide deployment. Besides, the intrinsic characteristics of IoDT such as decentralized structure, information-centric routing and semantic communications entail critical challenges for security service provisioning in IoDT. To this end, this paper presents an in-depth review of the IoDT with respect to system architecture, enabling technologies, and security/privacy issues. Specifically, we first explore a novel distributed IoDT architecture with cyber-physical interactions and discuss its key characteristics and communication modes. Afterward, we investigate the taxonomy of security and privacy threats in IoDT, discuss the key research challenges, and review the state-of-the-art defense approaches. Finally, we point out the new trends and open research directions related to IoDT.