The demand for using shell theories comes from its efficiency in computational and analytical cost. On another side, new materials that are orthotropic and/or anisotropic in nature are discovered and broadly used in many fields. Many advanced shell theories are developed for these new materials, particularly in the recent decades. A study about the accuracy of these shell theories is very meaningful to build confidence in them for further applications. This study requires a precise benchmark against which shell theories can be tested. This is the main research subjective in this dissertation: to build a set of solutions using the three dimensional (3D) theory of elasticity against which shell theories can be tested for accuracy. The contents of this dissertation to support this research include a comprehensive literature review for the shell theories and recent usage and to find the gaps which need to be filled. These gaps include, among others, the lack of studies on the accuracy of the theories used and the absence of results using the 3D theory, particularly for orthotropic materials. Some of these studies are conducted here. The deficiency of some commercial finite element packages is discussed here. The reasons for the absence of accurate results are investigated. The 3D theory and analyses of isotropic and orthotropic materials of hollow cylinders is investigated here for reliable results.