Cylindrical shells are fundamental and core structural elements in various fields of engineering and technology. They have been extensively used for the purpose of load- carrying in the past. Recently they have acquired a paramount and valuable place in practical applications constructing gas cylinders, pressure vessels, boilers, pipe lines, arch dams, submarines, tunnels, missiles and so on. Prior to their practical uses, their different dynamical aspects, such as vibration, buckling, and stability have been studied analytically to avoid any fatigue and risk that may implicate a physical system. Particularly, the analyses of their vibration characteristics have attracted the attention of theoretical researchers in recent years for their vital uses. In this dissertation, vibrations of cylindrical shells are investigated by involving Winkler and Pasternak elastic foundations. Fluid loading terms are also introduced to examine the influence of fluids on the shell vibrations. The present shells are considered to be constructed from isotropic and functionally graded materials. The vibrations of functionally graded cylindrical shells with exponential volume fraction law are investigated. Shell frequencies are varied by changing the bases of the law. The shell problem is solved by employing the Rayleigh-Ritz method for the simply supported boundary conditions. It is found that both the values of bases and exponents of the volume fraction law influence the frequencies of the shell. The vibration characteristics of functionally graded cylindrical shells based on the Winkler and Pasternak foundations are investigated. The Wave Propagation approach has been utilized to solve the shell dynamical equations. It is found that the influence of these elastic foundations is more pronounced for the circumferential wave numbers (n). Shell frequencies increase as the circumferential wave number is increased. Also the vibration analysis of fluid-filled cylindrical shells as well as fluid-filled functionally graded cylindrical shells resting on elastic foundations is examined. It is noted that the fluid-loading terms reduce frequencies of fluid-filled cylindrical shells to almost up to half of their corresponding values for empty cylindrical shells. It is concluded that the natural frequencies of the functionally graded cylindrical shells are greatly affected when the shells are filled with fluid and placed on elastic foundations. Three research papers of the author have been published in internationally recognized journals and one is accepted.