Texture evolution is a vital component of many computational tools that link structure, properties and processes of polycrystalline materials. By definition, this evolution process involves the manipulation, via rotation, of points in orientation space. The computational requirements of the current methods being used to rotate crystalline orientations are a significant limiting factor in the drive to merge the texture information of materials into the engineering design process. The goal of this research is to find and implement a practical rotation algorithm that can significantly decrease the computation time required to rotate macroscopic and microscopic crystallographic textures. Three possible algorithms are considered in an effort to improve the computational efficiency and speed of the rotation process. The first method, which will be referred to as the Gel'fand method, is based on a paper, [1], that suggests a practical application of some of Gel'fand's theories for rotations [2]. The second method, which will be known as the streamline method, is a variation on the Gel'fand method. The third method will be known as the principal orientation method. In this method, orientations in Fourier space are written as linear combinations of points on the convex surface of the microstructure hull to reduce the number of points that must be rotated during each step in the texture evolution process. This thesis will discuss each of these methods, their strengths and weaknesses, and the accuracy of the computational results obtained from their implementation.