Person re-identification is a very challenging research topic in the field of computer vision. In recent years, with the emergence of large-scale datasets and development of deep learning, many results have been obtained in the study of person feature extraction and description, distance metric learning. In the existing literature, feature extraction and description have been well summarized, but there is no comprehensive analysis of metric learning. Considering the key role of metric learning in improving person re-identification, it is necessary to systematically review the research status of metric learning in person re-identification. This survey gives a systematic summary of metric learning methods from three perspectives: distance metric methods, metric learning algorithms and re-ranking algorithms. Then, the performance of some representative methods are compared and analyzed. Finally, we make a prospect for the future research direction of metric learning.