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The equivariant degree theory was introduced in beginning of 90s, by Ize, Vignoli, Massabo and independently by Geba, Krawcewicz, and Wu, as an alternative to the equivariant singularity theory for studying symmetric properties of solutions and symmetric bifurcation problems for various classes of differential equations. I will discuss the ideas behind the construction of the equivariant degree theory and the most recent developments of new theoretical results and their implementation into qualitative methods for studying the existence of periodic or quasi-periodic solutions in dynamical systems. Several examples illustrating applications of these methods and computational aspects of the equivariant degree will be presented.