From the cooling of the early universe to quenching in magnets, driving a system through a continuous phase transition results in fascinating non-equilibrium dynamics. Close to the critical point separating two phases of matter, the intrinsic relaxational timescale diverges and no matter how slow the drive, the system is thrown out of equilibrium to produce defects. I will introduce this phenomenon in its original context of structure formation, equally applicable the early universe and to condensed states of matter, such as liquid Helium, and describe related universal ‘Kibble-Zurek’ physics. I will then explore the tantalizing way in which the dynamics plays in quantum phases of matter, in particular, quantum magnets and a class of systems that has recently stolen the limelight – topological materials. Finally, I will return to the cosmic realm by drawing parallels between quantum dynamics in topological systems and classical dynamics in black holes.