Mobile Edge Computing (MEC) brings the benefits of cloud computing, such as computation, networking, and storage resources, close to end users, thus reducing end-to-end latency and enabling various novel use cases, such as vehicle platooning, autonomous driving, and the tactile internet. However, frequent user mobility makes it challenging for the MEC to guarantee the close proximity to the users. To tackle this challenge, the underlying network has to be capable of seamlessly migrating applications between multiple MEC sites. This application migration requires the quick and flexible migration of the application states without service interruption, while minimizing the state transfer cost. In this article, we first study the state transfer optimization problem in the MEC. To solve this problem, we propose a metaheuristic algorithm based on Tabu search. We then propose Flexible and Low-Latency State Transfer in Mobile Edge Computing (FAST), the first programmable state forwarding framework. FAST flexibly and directly forwards states between source instance and destination instance based on Software-Defined Networking (SDN). Both simulation results and practical testbed results demonstrate the favorable performance of the proposed Tabu search algorithm and the FAST framework compared to the state-of-the-art schemes.