Abstract : There is a rich interplay between the degree of regularity of a group action on the circle and the allowable algebraic structure of the group. In this series of talks, I will outline some highlights of this theory, culminating in a construction due to Kim and myself of groups of every possible critical regularity $\alpha \in [1,\infty)$.There is a rich interplay between the degree of regularity of a group action on the circle and the allowable algebraic structure of the group. In this series of talks, I will outline some highlights of this theory, culminating in a construction due to Kim and myself of groups of every possible critical regularity $\alpha \in [1,\infty)$.