Renormalization is the technique of analyzing a dynamical system by understanding its small-scale behavior. This approach has proved to be very powerful, and has produced many deep results that would have been inaccessible through more conventional methods. In my talk, I will motivate the use of renormalization in the study of holomorphic dynamical systems with a rotation domain (called a Siegel disk). The focus will be in the two-dimensional setting, where many of the essential tools of one-dimensional holomorphic dynamics are not available. As the main result, I will prove that the boundary of a golden-mean Siegel disks in two-dimensions is typically not a quasi-circle.