5G systems are envisioned to support numerous delay-sensitive applications such as the tactile Internet, mobile gaming, and augmented reality. Such applications impose new demands on service providers in terms of the quality of service (QoS) provided to the end-users. Achieving these demands in mobile 5G-enabled networks represent a technical and administrative challenge. One of the solutions proposed is to provide cloud computing capabilities at the edge of the network. In such vision, services are cloudified and encapsulated within the virtual machines or containers placed in cloud hosts at the network access layer. To enable ultrashort processing times and immediate service response, fast instantiation, and migration of service instances between edge nodes are mandatory to cope with the consequences of user’s mobility. This paper surveys the techniques proposed for service migration at the edge of the network. We focus on QoS-aware service instantiation and migration approaches, comparing the mechanisms followed and emphasizing their advantages and disadvantages. Then, we highlight the open research challenges still left unhandled.