Person search has drawn increasing attention due to its real-world applications and research significance. Person search aims to find a probe person in a gallery of scene images with a wide range of applications, such as criminals search, multicamera tracking, missing person search, etc. Early person search works focused on image-based person search, which uses person image as the search query. Text-based person search is another major person search category that uses free-form natural language as the search query. Person search is challenging, and corresponding solutions are diverse and complex. Therefore, systematic surveys on this topic are essential. This paper surveyed the recent works on image-based and text-based person search from the perspective of challenges and solutions. Specifically, we provide a brief analysis of highly influential person search methods considering the three significant challenges: the discriminative person features, the query-person gap, and the detection-identification inconsistency. We summarise and compare evaluation results. Finally, we discuss open issues and some promising future research directions.