With the advances of the Internet of Things (IoT) and 5G/6G wireless communications, the paradigms of mobile computing have developed dramatically in recent years, from centralized mobile cloud computing to distributed fog computing and mobile edge computing (MEC). MEC pushes compute-intensive assignments to the edge of the network and brings resources as close to the endpoints as possible, addressing the shortcomings of mobile devices with regard to storage space, resource optimisation, computational performance and efficiency. Compared to cloud computing, as the distributed and closer infrastructure, the convergence of MEC with other emerging technologies, including the Metaverse, 6G wireless communications, artificial intelligence (AI), and blockchain, also solves the problems of network resource allocation, more network load as well as latency requirements. Accordingly, this paper investigates the computational paradigms used to meet the stringent requirements of modern applications. The application scenarios of MEC in mobile augmented reality (MAR) are provided. Furthermore, this survey presents the motivation of MEC-based Metaverse and introduces the applications of MEC to the Metaverse. Particular emphasis is given on a set of technical fusions mentioned above, e.g., 6G with MEC paradigm, MEC strengthened by blockchain, etc.