In 2000, Frame and Cogevina introduced a method for constructing fractals using circle inversion maps. The literature focuses on the graphical aspect of such fractals, without presenting a careful development of the underlying mathematical framework. In this thesis, we present such a framework, making a strong connection to iterated function systems (IFS) theory. Our final result establishes that the set valued system of circle inversion maps induced by a collection of possibly overlapping circles in the plane has a unique set attractor. We then establish a similar mathematical framework in the setting of non-touching star-shaped sets. We present graphical examples for both settings using the chaos game. Finally, fractals literature develops the well-known concept of local iterated function systems with grey-level maps, with applications to image processing. We follow this path to establish a framework that uses local circle inversion maps as the functions. We demonstrate the results with examples. Ontario Graduate Scholarship