The aim of the book is the development and applications of the degree theory in the context of equivariant maps. (Equivariant simply means that the mapping has certain symmetries, e.g., being even/odd, periodic, rotational invariant, etc.). The theory is developed both in finite and infinite dimension. The first chapter gives necessary preliminaries. The second chapter brings the definition of the degree and studies its basic properties. As the definition is somewhat abstract (the degree is defined as an element of the group of equivariant homotopy classes of maps between two spheres), it is useful to compute the degree in various particular cases. This is accomplished in Chapter 3. The last and also the longest chapter, deals with applications to particular ODE’s and to bifurcation theory. The aim of the authors was to write a book that would be easily accessible even to non-specialists, thus the exposition is accompanied by a number of examples and the use of abstract special tools is limited. It is also worth noting that each chapter is accompanied by detailed bibliographical remarks.