Abstract: ‍ ‍Recently， direction of arrival （DOA） estimation using sparse arrays has emerged as one of the prominent topics in the field of array signal processing. Compared with traditional uniform linear arrays， sparse arrays have attracted extensive attention and systematic academic research due to their exceptional properties， such as large aperture， increased degrees of freedom， alleviated mutual coupling， reduced redundancy， low overhead， and flexible deployment. Meanwhile， to completely leverage the immense advantages offered by sparse arrays， scholars have developed a series of DOA estimation algorithms from various perspectives to further enhance the number of resolvable sources and improve angle estimation accuracy. In this paper， we provide an elaborate account of the historical development and representative achievements in both sparse array design and DOA estimation algorithms by constructing the sparse array signal model and clarifying the related terminologies. In the aspect of sparse array design， the concepts of different types of sparse arrays were deeply analyzed around the core indicators， including degrees of freedom， mutual coupling， and redundancy. In particular， we focus on two classes of structured sparse arrays， i.e.， nested arrays and coprime arrays， and highlight the number of their consecutive degrees of freedom and degrees of freedom. In terms of sparse array DOA estimation， we expound two types of direction-finding theories based on physical array processing and virtual array processing according to the different construction principles of signal parameters， and the applicable conditions and performance advantages associated with each method are thoroughly analyzed. Furthermore， the Cramér-Rao bound （CRB） for DOA estimation with sparse arrays is also reviewed， which serves an important benchmark for evaluating the pros and cons of distinct arrays and algorithms. Finally， we forecast the future directions by analyzing the problems of existing achievements， aiming to provide a theoretical foundation and technical assistance for the engineering application of sparse array DOA estimation.