Texture (preferred grain orientation) of polycrystalline metal sheets is an important material property which influences the process of making products such as aluminum cans and automobile and aircraft components. Traditionally, the texture is determined by X-ray or neutron diffraction techniques which are often destructive and time-consuming in nature. Over the last decade, ultrasonics has been found to provide an alternative way to characterize texture nondestructively and quickly. This dissertation makes an in-depth study of the fundamental physical principles of ultrasonic characterization of texture in metal sheets of cubic and hexagonal crystallites. The foundation is laid by investigations of wave propagation in anisotropic plates. This is then used to evaluate the accuracy of previously developed, conventional ultrasonic techniques for texture characterization. Through these studies, improvement of the conventional techniques has been made and new ultrasonic techniques have been proposed and applied to some Al samples. This dissertation also presents the extension of these techniques to characterization of the texture of hexagonal polycrystallites such as Ti and Zr. Discussions of ultrasonic velocity measurement methods and errors are also included.