In the past few years our understanding of CFTs in higher dimensions has dramatically advanced. General structural properties have been clarified, many new (supersymmetric) theories have been constructed, and powerful new tools have been developed. A surprisingly versatile and effective tool is the modern incarnation of the conformal bootstrap. In this talk I will illustrate some of these new ideas focussing on N=2 and N=4 superconformal field theories in four dimensions. I will outline the (complete?) catalogue of these theories that has emerged in recent years and describe what we are learning about their operator spectrum.
