I will review some recent progress on six-dimensional conformal field theories, highlighting applications to thedynamics of M5-branes. On an orbifold singularity, M5s can fractionate, and they can recombine by sometimesleaving behind a 'frozen' version of the singularity. The CFTs describing these processes also happen to beubiquitous building blocks for more complicated ones. There is also evidence for other processes where branesrecombine in patterns associated to nilpotent orbits of Lie groups, with precise checks obtained by anomalies andmoduli space dimensions.