Human motion synthesis is a fundamental task in the field of computer animation. Recent methods based on diffusion models or GPT structure demonstrate commendable performance but exhibit drawbacks in terms of slow sampling speeds or the accumulation of errors. In this paper, we propose Motion Flow Matching, a novel generative model designed for human motion generation featuring efficient sampling and effectiveness in motion editing applications. Our method reduces the sampling complexity from 1000 steps in previous diffusion models to just 10 steps, while achieving comparable performance in text-to-motion and action-to-motion generation benchmarks. Noticeably, our approach establishes a new state-of-the-art result of Fréchet Inception Distance on the KIT-ML dataset. What is more, we tailor a straightforward motion editing paradigm named trajectory rewriting leveraging the ODE-style generative models and apply it to various editing scenarios including motion prediction, motion in-between prediction, motion interpolation, and upper-body editing.