Abstract: As an emerging research field bridging geometry and architectural design, architectural geometry focuses on geometric issues across various stages of contemporary architectural design and fabrication. Its academic scope spans multiple disciplines, including architecture, applied geometry, computational design, and fabrication technologies. Discrete surface modeling, especially quadrilateral-based discrete modeling, is one of the core methods in architectural geometry and significantly impacts the design and construction process. Compared to traditional triangular meshes, quadrilateral meshes offer significant advantages in parametric modeling, structural design, and construction feasibility analysis. This paper systematically reviews the research progress related to quadrilateral structures in architectural geometry, focusing on recent advancements in discrete surface modeling, self-supporting structures, and fabrication-aware design. The paper explores the important role of quadrilateral meshes in structural engineering, particularly in the geometry design for construction. It also discusses future development trends and aims to provide a valuable reference for subsequent research in this field.