The Metaverse has emerged as the next generation of the Internet. It aims to provide an immersive, persistent virtual space where people can live, learn, work and interact with each other. However, the existing technology is inadequate to guarantee high visual quality and ultra-low latency service for the Metaverse players. Mobile Edge Computing (MEC) is a paradigm where proximal edge servers are utilized to perform computation-intensive and latency-sensitive tasks like image processing and video analysis. In MEC, the large amount of data is processed by edge servers closest to where it is captured, thus significantly reducing the latency and providing almost real-time performance. In this paper, we integrate fundamental elements (5G and 6G wireless communications, Blockchain, digital twin and artificial intelligence) into the MEC framework to facilitate the Metaverse. We also elaborate on the research problems and applications in the MEC-enabled Metaverse. Finally, we provide a case study to establish a thorough knowledge of the user utility maximization problem in a real-world scenario and gain some insights about trends in potential research directions.