The next generation (6G) wireless systems aim to cater to the Internet of Everything (IoE) and revolutionize customer services and applications to a fully intelligent and autonomous system. To achieve this, the digital twin edge network (DITEN) is proposed to combine mobile/multi-access edge computing (MEC) and digital twin (DT), thereby improving the network performance such as throughput and security, and reducing the cost of communication, computation, and caching. In DITENs, the network status can be continuously monitored, and based on the obtained network states, the networking schemes, such as routing and resource management, can be studied in the established DITENs from a centralized perspective. In this survey, we present a comprehensive overview of DITEN for 6G. First, we present the fundamental aspects of DITEN, including concept, framework, and potential. Second, a comprehensive design of DITEN is devised, including the DT modeling/updating, DT deployment, key issues, and enabling technologies. Then, the typical applications of DITEN towards 6G are provided, including the Internet of Things (IoT), vehicular network, spaceair-groud integrated network (SAGIN), healthcare, wireless systems, and other applications, along with the design of DITEN in each application, such as DT modeling, DT association, incentive mechanisms, and so on. Finally, challenges and open issues are discussed.