Digital twin (DT) is the recurrent and common feature in discussions about future technologies, bringing together advanced communication, computation, and artificial intelligence, to name a few. In the context of Industry 4.0, industries such as manufacturing, automotive, and healthcare are rapidly adopting DT-based development. The main challenges to date have been the high demands on communication and computing resources, as well as privacy and security concerns, arising from the large volumes of data exchanges. To achieve low latency and high security services in the emerging DT, multi-tier computing has been proposed by combining edge/fog computing and cloud computing. Specifically, low latency data transmission, efficient resource allocation, and validated security strategies of multi-tier computing systems are used to solve the operational problems of the DT system. In this paper, we introduce the architecture and applications of DT using examples from manufacturing, the Internet-of-Vehicles and healthcare. At the same time, the architecture and technology of multi-tier computing systems are studied to support DT. This paper will provide valuable reference and guidance for the theory, algorithms, and applications in collaborative multi-tier computing and DT.