Mobile Edge Computing (MEC) is a new network architecture concept. Aiming to mitigate the root cause of latency as well as long distance between devices and central cloud, MEC brings the computation and storage resources as close as possible to end devices. However, mobility of end devices brings challenging issues for service continuity and Quality of Service (QoS) of these systems. In this paper, the state-of-the-art research efforts towards handling user mobility in a MEC environment for different kinds of services are surveyed. Also, open research problems are discussed.