Video streaming market has been growing rapidly for the past two decades. YouTube and Netflix, for example, are part of everyday life for many of us. Although network bandwidth has increased with the development of 5G and fiber, user demand and expectations have also increased with 4K videos, larger screens, and COVID-19. Recent explosive increase in video traffic made it difficult for the network and applications to satisfy the user requirements of video streaming. Multi-access edge computing (MEC) is a promising technology to resolve this problem. MEC is a network architecture concept that offers cloud-computing capabilities and an IT service environment at the edge of the network. In this paper, we review the research efforts in video streaming with MEC over the past few years. We have clustered and categorized the literature according to the optimization criteria, with focus on cache management. We aim this survey to be a cornerstone for designing effective novel caching scheme in MEC for video streaming.