Person re-identification (Re-ID) has been a significant research topic in the past decade due to its real-world applications and research significance. While supervised person Re-ID methods achieve superior performance over unsupervised counterparts, they can not scale to large unlabelled datasets and new domains due to the prohibitive labelling cost. Therefore, unsupervised person Re-ID has drawn increasing attention for its potential to address the scalability issue in person Re-ID. Unsupervised person Re-ID is challenging primarily due to lacking identity labels to supervise person feature learning. The corresponding solutions are diverse and complex, with various merits and limitations. Therefore, comprehensive surveys on this topic are essential to summarise challenges and solutions to foster future research. Existing person Re-ID surveys have focused on supervised methods from classifications and applications but lack detailed discussion on how the person Re-ID solutions address the underlying challenges. This survey review recent works on unsupervised person Re-ID from the perspective of challenges and solutions. Specifically, we provide an in-depth analysis of highly influential methods considering the four significant challenges in unsupervised person Re-ID: 1) lacking ground-truth identity labels to supervise person feature learning; 2) learning discriminative person features with pseudo-supervision; 3) learning cross-camera invariant person feature, and 4) the domain shift between datasets. We summarise and analyse evaluation results and provide insights on the effectiveness of the solutions. Finally, we discuss open issues and suggest some promising future research directions.