Abstract: The cylindrical shell structures is the fundamental component widely used in the engineering field. Its structural stability under external loads is not only a challenging topic in the study of plate and shell theory but also a key issue of great concern in engineering practice. This paper focuses on the critical buckling problem of cylindrical shell structures and systematically elaborates on the governing equations of stability and analytical methods under various loading conditions. Firstly, the classical linear buckling theory and its governing equations are introduced. Subsequently, the differential equations that describe the critical buckling behavior of thin cylindrical shell structures under uniform axial compression and non-uniform axial body force loads are derived in detail. The theoretical solution process for eigenvalue buckling analysis is also provided. Through the discussion in this paper, students can obtain a deeper understanding of the critical load solution program and the stability of cylindrical shells, offering theoretical guidance for related engineering design and safety assessment.