Emerging cloud-centric networks span from edge clouds to large-scale datacenters with shared infrastructure among multiple tenants and applications with high availability, isolation, fault tolerance, security, and energy efficiency demands. Live migration (LiMi) plays an increasingly critical role in these environments by enabling seamless application mobility covering the edge-to-cloud continuum and maintaining these requirements. This survey presents a comprehensive survey of recent advancements that democratize LiMi, making it more applicable to a broader range of scenarios and network environments both for virtual machines (VMs) and containers, and analyzes LiMi’s technical underpinnings and optimization techniques. It also delves into the issue of connections handover, presenting a taxonomy to categorize methods of traffic redirection synthesized from the existing literature. Finally, it identifies technical challenges and paves the way for future research directions in this key technology.