The concept of points within a rigid body are used to firstly identify the minimum number of parameters required to represent a rotation in three-dimensional space. The approach taken is the same as that used in Goldstein’s book Classical Mechanics [GPS01]. This is followed by the development of the concepts of a body (local) frame and a reference frame, which are used to construct a Direction Cosine Matrix (DCM), also known as a rotation matrix, to represent and perform a rotation. The rotation matrix is used for both passive and active rotations. In this chapter, passive rotations are illustrated using frame rotations, while active rotations are shown using vector rotations. The properties of the rotation matrix as an orthogonal transformation are obtained and its role in the change of basis using the similarity transformation is described.