Metaverse is an emerging virtual universe where humans can have real-time interactions and solid social links like in the physical world, and it opens up a new era of Internet and interactions. In Metaverse, an immersive and photorealistic environment promotes social activities, including education, meetings, and shopping of digital avatars based on critical technologies, including 3D rendering, extended reality, digital twins, artificial intelligence, and Blockchain. However, the limitations of computation, storage, and energy resources restrict the development of Metaverse, and a series of system issues (e.g., latency, security, and battery-life) continue to arise. As a result, how to find corresponding measurements to mitigate unsatisfactory influences becomes the focus. Mobile edge computing (MEC) as a distributed computing paradigm offloads computation-intensive tasks to the edge of the network. It brings the resources as close as possible to the end devices, addressing the shortcomings mentioned above. In this paper, we propose a comprehensive survey of the MEC-based Metaverse. Particular emphasis is given to the technologies convergence, architectures, and application scenarios, e.g., BoundlessXR and CloudXR. Significantly, we introduce the potential future directions for developing Metaverse systems.