A comprehensive introduction to bifurcation theory in the presence of symmetry, an applied mathematical topic which has developed considerably since the late 1970s. The book has two aims. One is to expound the mathematical methods of equivariant bifurcation theory. Beyond the classical bifurcation tools, such as centre manifold and normal form reductions, the presence of symmetry requires the introduction of the algebraic and geometric formalism of Lie group theory and transformation group methods. The other aim is to present the most recent ideas and results in this theory, in relation to applications. This includes bifurcation of relative equilibria and relative periodic orbits for compact and noncompact group actions, heteroclinic cycles and forced symmetry-breaking perturbations. Although not all recent contributions could be included and a choice had to be made, a rather complete description of these developments is provided. At the end of every chapter, exercises are offered to the reader.