Group-theoretic bifurcation theory is introduced as a means to describe qualitative aspects of symmetry-breaking bifurcation, such as possible types of critical points and the symmetry of bifurcating solutions. We advance a series of mathematical concepts and tools, including: group equivariance, Liapunov–Schmidt reduction, equivariant branching lemma, and block-diagonalization. The theory of linear representations of finite groups in Chap. 7 forms a foundation of this chapter. This chapter is an extension of Chap. 2 to a system with symmetry and a prerequisite to the study of structures and materials with dihedral symmetry in Chaps. 9 – 13 and larger symmetries in Chaps. 14 – 17.