Shells of double curvature are common structural elements in aerospace and related industries, but due to the complexity of their configurations and governing equations, little has been done to classify their general dynamic behavior. The subject of this dissertation is the determination of the effect of the meridional curvature on the natural vibrations of a class of axisymmetrically prestressed doubly curved shells of revolution. A set of linear equations governing the infinitesimal vibrations of axisymmetrically prestressed shells is developed from Sander's nonlinear shell theory and both the in-plane inertia and prestress deformation effects are retained in the development. The equations derived are consistent with first-order thin-shell theory and can be used to describe the behavior of shells with arbitrary meridional configuration having moderately small prestress rotations. A numerical procedure is given for solving the governing equations for the natural frequencies and associated mode shapes for a general shell of revolution with homogeneous boundary conditions. The numerical procedure uses matrix methods in finite-difference form coupled with a Gaussian elimination to solve the governing eigenvalue problem. An approximate set of governing equations of motion with constant coefficients which are based on shallowness of the meridian are developed as an alternate more rapid method of solution and are solved in an exact manner for all boundary conditions. The solutions of the exact system of shell equations determined from the numerical procedure are used to determine the accuracy of the approximate solutions and with its accuracy established, the approximate equations are used exclusively to generate results. The membrane and pure bending equations which correspond to the approximate set of equations are solved for a specific boundary condition. The effect of the meridional curvature on the fundamental frequencies of a class of cylindrical-like shells with shallow meridional curvature and freely supported edges are investigated. Results show that the positive Gaussian curvature shells have fundamental frequencies well above those of corresponding cylindrical shells. The fundamental frequencies of the negative Gaussian curvature shells generally are below those of the corresponding cylinders and evidence wide variations in value with large reductions in magnitude occuring at certain critical curvatures. Comparison of the membrane, pure bending and complete shell analyses shows that these critical curvatures represent configurations at which the fundamental mode of vibration of the shell is in a state close to pure bending. The membrane theory affords a simple method of determining the modal wavelength ratio at which the pure bending state exists for a given negative Gaussian curvature shell, while the pure bending theory gives a good estimate of the magnitude of the frequency for this wavelength ratio. Meridional edge restraints and internal lateral pressure reduce the wide variation of the natural frequencies in the negative curvature shells and in general raise the natural frequencies. External lateral pressure accentuates the reduction in natural frequencies of the negative curvature shells and causes instability at low compressive stress ratios. / Ph. D.