Basic Concepts. Introduction. Transformation properties of tensors. Description of material symmetry. Restrictions due to material symmetry. Constitutive equations. Group Representation Theory. Introduction. Elements of group theory. Group representations. Schur's Lemma and orthogonality properties. Group characters. Continuous groups. Elements of Invariant Theory. Introduction. Some fundamental theorems. Invariant Tensors. Introduction. Decomposition of property tensors. Frames, standard tableaux and young symmetry operators. Physical tensors of symmetry class (n1 n2...). The inner product of property tensors and physical tensors. Symmetry class of products of physical tensors. Symmetry types of complete sets of property tensors. Examples. Character tables for symmetric groups S2, ... , S8. Group Averaging Methods. Introduction. Averaging procedure for scalar-valued functions. Decomposition of physical tensors. Averaging procedures for tensor-valued functions. Examples. Generation of property tensors. Anisotropic Constitutive Equations and Schur's Lemma. Introduction. Application of Schur's Lemma: Finite groups. The crystal class D3. Product tables. The crystal class S4. The transversely isotropic groups T1 and T2. Generation of Integrity Bases: The Crystallographic Groups. Introduction. Reduction to standard form. Integrity bases for the triclinic, monoclinic, rhombic, tetragonal and hexagonal crystal classes. Invariant functions of a symmetric second-order tensor: C3. Generation of product tables. Generation of Integrity Bases: Continuous Groups. Introduction. Identities relating 3 x 3 matrices. The Rivlin-Spencer procedure. Invariants of symmetry type (n1 ... np. Generation of the multilinear elements of an integrity basis. Computation of Pn, Pn1 ...np, Qn, Qn1... np. Invariant functions of traceless symmetric second-order tensors: R3. An integrity basis for functions of skew-symmetric second-order tensors and traceless symmetric second-order tensors: R3. An integrity basis for functions of vectors and traceless symmetric second-order tensors: O3. Transversely isotropic functions. Generation of Integrity Bases: The Cubic Crystallographic Groups. Introduction. Tetartoidal class, T, 23. Diploidal class, Th, m3. Gyrodial class, 0, 432: Hextetrahedral class, Td, 43m. Hexoctahedral class, Oh, m3m. Irreducible Polynomial Constitutive Expressions. Introduction. Generating functions. Irreducible expressions: The crystallographic groups. Irreducible expressions: The orthogonal groups R3, 03. Scalar-valued invariant functions of a traceless symmetric third-order tensor F: R3, O3. Scalar-valued invariant functions of a traceless symmetric fourth-order tensor V: R3. References. Index.