Since FGM orthotropic structures have such striking qualities as high strength, exceptional stiffness, stiffness-to-weight ratio, reduced cost, and high strength-to-weight ratio, they are employed extensively in the mechanical, aerospace, and civil engineering sectors. Thick plates and shells have more noticeable shear deformation effects. Therefore, in recent years, there has been a lot of interest in the vibration and buckling investigation of FGMs orthotropic plates and shells. Moreover, researchers have developed a variety of approaches and procedures for the examination of orthotropic FGM plates and shells. The majority of the literature review in this publication is focused on orthotropic FGMs plate and shell buckling and linear and nonlinear free vibration. In engineering practices, it is customary to use material-oriented or orthotropic materials in several domains to optimize the structures and maximize material properties, which is especially crucial for FG constructions. Solutions for the orthotropic FGM structure are studied analytically and numerically with different plate and shell theories.