The subject of free vibration analysis of thin cylindrical shells is one that extends well back into the last century. In general, the computational methods can be classified as analytical methods and numerical methods. Based on the Flugge thin shell theory, this thesis presents exact solutions for vibration of closed and open cylindrical shells. The state-space technique is adopted to derive the homogenous differential equations for a shell segment and the domain decomposition method is employed to impose the equilibrium and compatability requirements along the interfaces of the shell segments. Extensive analytical and numerical results have been obtained in this thesis for vibration of open/closed cylindrical shells with different boundary conditions, step-wise thickness variations, and multiple intermediate ring supports. The results can serve as useful benchmark values for researchers and engineers to validate their numerical method for shell analysis. / Doctor of Philosophy (PhD)