Abstract: From the cooling of the early universe to quenching in magnets, driving a system through a continuous phase transition results in fascinating 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. Such non-equilibrium behavior was first brought to attention in the context of domain formation in the early universe by Kibble and in that of condensed matter systems, such as superfluid Helium, by Zurek. While the original Kibble-Zurek scenarios involve dialing down or “quenching” the temperature, analogs involving quantum quenching between quantum phases are gaining prominence. Here, a simple picture of this Kibble-Zurek scenario and its application in a magnetic system will be presented. We will then explore how this tantalizing dynamics plays out in a class of systems that have recently stolen the limelight – topological materials.