Seeking reliable correspondences between two feature sets is a fundamental and important task in computer vision. This paper attempts to remove mismatches from given putative image feature correspondences. To achieve the goal, an efficient approach, termed as locality preserving matching (LPM), is designed, the principle of which is to maintain the local neighborhood structures of those potential true matches. We formulate the problem into a mathematical model, and derive a closed-form solution with linearithmic time and linear space complexities. More specifically, our method can accomplish the mismatch removal from thousands of putative correspondences in only a few milliseconds. Experiments on various real image pairs for general feature matching, as well as for visual homing and image retrieval demonstrate the generality of our method for handling different types of image deformations, and it is more than two orders of magnitude faster than state-of-the-art methods in the same range of or better accuracy.