The prevalence of computation-intensive and latency-sensitive mobile applications, such as mobile augmented reality (MAR) and autonomous driving, has an utmost effect on resource-limited mobile clients. Mobile edge computing (MEC) is proposed to be a promising paradigm to bridge the gap between the stringent computation and latency requirements of mobile applications and the constrained computation and battery capacity on mobile clients. The main feature of MEC is to push mobile computing, network control and storage to the network edges (e.g., base stations (BSs) and access points (APs)). However, prior work on MEC fails to achieve their expected performance in multiple practical cases, e.g., irreparable network disruptions caused by wireless link instability or user-mobility that is a critical characteristic of mobile applications.In this dissertation, fast and energy-efficient mobility management in MEC networks is explored. Link-instability and user-mobility incurred challenges in MEC are addressed from four steps. (1) An intelligent handoff trigger mechanism is designed to achieve a fast and accurate trigger for seamless mobility support in MEC networks. (2) Fast and energy-efficient radio-service handoff protocols are established in order to rebuild offloading services on a new MEC server with low overhead after a handoff is triggered at a mobile client in MEC networks. (3) To minimize performance degradation during mobility caused by radio resource allocation unfairness, single and multiple edge server radio resource allocation protocols to impartially allocate the uplink and the downlink radio resources in MEC networks are proposed. (4) A dynamic configuration adaptation algorithm is proposed for mobile clients to achieve energy-efficient offloading in MEC networks while satisfying variant clients' user preferences. In summary, this research is essential for providing fast and energy-efficient mobility support for mobile clients in MEC networks. In addition, this research provides critical insights for future designs of mobility management in MEC networks.