The topic of this graduate thesis is one of lesser known gems of elementary geometry, The six circles theorem, and some of its generalizations. This theorem is one of many theorems in the book ”The seven circles theorem and other new theorems” which is a result of collaboration of three geometry enthusiasts, C. J. A. Evelyn, G. B. Money-Coutts, and J. A. Tyrrell. The first chapter contains a short history of The six circles theorem’s origins and two different forms and proofs of it. The theorem concerns chains of circles inscribed in a given triangle \(P_{1}P_{2}P_{3}\): the first circle, \(C_{1}\), inscribed in the first angle at \(P_{1}; C_{2}\), inscribed in the angle at \(P_{2}\) and tangent to the circle \(C_{1} ; C_{3}\), inscribed in the angle at \(P_{3}\) and tangent to \(C_{2}\), and so on, cyclically. The claim of the theorem is that this process is periodic, that is, \(C_{7}=C_{1}\). The first form of the theorem holds for a chain of circles for which all tangency points lie on the sides of the triangle, and not their extensions. Secondly, chains of circles are observed for which the next circle can be tangent to a side of the triangle but also to its extension. It is proven that, in general, the chain is eventually 6-periodic but may have an arbitrarily long pre-period. In the second chapter it is investigated whether The six circles theorem extends to polygons other than triangles. It is shown that there is a subclass of irregular \(n\)-gons for which periodicity holds. Finally, an interesting variation of the main theorem is considered, where the sides of a triangle are replaced by circles. It was proven by Tyrrell and Powell that the 6-periodicity persists even then.