With the advent of fog and edge computing paradigms, computation capabilities have been moved toward the edge of the network to support the requirements of highly demanding services. To ensure that the quality of such services is still met in the event of users’ mobility, migrating services across different computing nodes becomes essential. Several studies have emerged recently to address service migration in different edge-centric research areas, including fog computing, multi-access edge computing (MEC), cloudlets, and vehicular clouds. Since existing surveys in this area focus on either VM migration in general or migration in a single research field (e.g., MEC), the objective of this survey is to bring together studies from different, yet related, edge-centric research fields while capturing the different facets they addressed. More specifically, we examine the diversity characterizing the landscape of migration scenarios at the edge, present an objective-driven taxonomy of the literature, and highlight contributions that rather focused on architectural design and implementation. Finally, we identify a list of gaps and research opportunities based on the observation of the current state of the literature. One such opportunity lies in joining efforts from both networking and computing research communities to facilitate future research in this area.