This minicourse is an introduction to the group of conformal transformations of the sphere. We will focus on its action on the space of non-singleton compact subsets of the sphere. 1. Conformal autosimilarity: we will see that the compact subsets whose orbit is closed are exactly the limit sets of convex cocompact subgroups. 2. Quasicircles: we will see that the compact subsets whose orbit closure contains only Jordan curves are exactly the quasicircles. 3. Minimal orbits: we will construct compact subsets of the sphere whose orbit is not closed but whose orbit closure is minimal.