Recently, with the development of the intelligent surveillance, person re-identification (Re-ID) has attracted lots of attention in the academic and industrial communities, which aims to associate person images of the same identity under different non-overlapping cameras. Most of the current research works focus on the supervised case where all given training samples have label information. Considering the high cost of data labeling, these methods designed for the supervised setting have poor generalization in practical applications. This study focuses on person re-identification algorithms under the weakly supervised case including the unsupervised case and the semi-supervised case and classify and describe several state-of-the-art methods. In the unsupervised setting, these methods are divided into five categories from different technology perspectives, which include the methods based on pseudo-label, image generation, instance classification, domain adaptation, and others. In the semi-supervised setting, these methods are divided into four categories according to the case discrepancy, which are the case where a small number of persons are labeled, the case where there are few labeled images for each person, the case based on tracklet learning, and the case where there are the intra-camera labels but no inter-camera label information. Finally, several benchmark person re-identification datasets are summarized and some experimental results of these weak-supervised person re-Identification algorithms are analyzed.