In order to study on the fuzzy C-means clustering algorithm deeply, starting from the angle of improving convergence speed of the algorithm, the membership correction fuzzy C-means clustering algorithms which are represented by RCFCM, S-FCM, PIM and FCMα algorithm etc. are summarized and the research progress is tracked. To show the panorama of the algorithms, the nature and characteristics of each algorithm are analyzed by the experiments with different parameters and different fuzzy index. According to the experimental results, the direction of further research of the algorithms is pointed out. The above work can provide a valuable reference for further research on FCM algorithm.