This book deals with necessary and sufficient conditions for the existence of axes and planes of symmetry. We discuss matrix representation of an elasticity tensor belonging to a trigonal, a tetragonal or a hexagonal material. The planes of symmetry of an anisotropic elastic material (if they exist) can be found by the Cowin-Mehrabadi theorem (1987) and the modified Cowin-Mehrabadi theorem proved by Ting (1996). Using the Cowin-Mehrabadi formalism Ahmad (2010) proved the result that an anisotropic material possesses a plane of symmetry if and only if the matrix associated with the material commutes with the matrix representing the elasticity tensor. A necessary and sufficient condition to determine an axis of symmetry of an anisotropic elastic material is given by Ahmad (2010). We review the Cowin-Mehrabadi theorem for an axis of symmetry and develop a straightforward way to find the matrix representation for a trigonal, a tetragonal or a hexagonal material.