The present thesis consists of two main parts. The first part is concerned with the vibration and statics of transversely isotropic thick-walled toroidal shells. The second part is concerned with the vibration and statics of orthotropic thin-walled toroidal shells. In the first part a solution based on the linear three-dimensional theory of elasticity is developed for vibration and static problems of toroidal shells. The theory is developed for transversely isotropic toroids of arbitrary but uniform thickness. In the semi-analytical method that is adopted Fourier series are written in the circumferential direction, forming a set of two-dimensional problems. Finally results are determined for local surface loading problems. In the second part a solution based on the linear elastic Sanders-Budiansky shell equations is developed. The vibration and static characteristics of orthotropic toroidal shells of variable thickness are considered. A semi-analytical method in which Fourier series are written in the circumferential direction is adopted, forming a set of one-dimensional problems. A novelty in the solution concerns the use of power series as trial functions in a domain exhibiting cyclic periodicity. Results are determined in the second part for two separate applications. The problems in both parts of the work are solved using the differential quadrature method. A commercial finite element program is used to determine alternative solutions. The results from these two methods are compared, and conclusions are drawn.