Abstract: Array design methods based on sparse optimization theory have attracted widespread attention because they can achieve high-precision beam pattern synthesis with fewer array elements while maintaining beam pattern resolution and sidelobe performance, as well as reducing mutual coupling effects and system costs. This paper provides a comprehensive overview of mainstream sparse linear array design methods and categorizes them into two classes: (i) grid-based sparse array design, including off-grid sparse array design methods; (ii) gridless sparse array design methods. From the perspectives of mathematical models, optimization algorithms, and performance analysis, the most representative algorithms for both single-pattern and multi-pattern design scenarios are introduced in detail, and the performance of different methods under various reference arrays, element constraints, and beam requirements is demonstrated. Finally, open problems in sparse array design and promising directions for future research are discussed.