Group actions are an efficient way of describing symmetries in objects by defining the essential elements of a given object as a set. The symmetries of the object are then defined as the symmetry group of this set. "Groups of Circle Diffeomorphisms" explores group actions on the simplest closed manifold, the circle. This text will be of interest to those doing research in several branches of mathematics, including dynamical systems, low-dimensional geometry and topology, and mathematical physics.