Michael Fisher once wrote that “the task of theory is to ... understand the universal aspects of the natural world.” This goal has motivated physicists in understanding phenomena from phase transitions to topological physics. In this talk, I will discuss how the quest for universality has progressed in an emerging field of research: the dynamics of coherent quantum systems. Dynamics plays a crucial role in ultracold AMO experiments and has recently become important in solid state systems such as superconducting circuits, where coherence is necessary for protecting and manipulating quantum information. I discuss canonical examples of universality in these systems, such as eigenstate thermalization, many-body localization, and the dynamics near phase transitions. Then I detail recent work where, in collaboration with two experimental groups, we demonstrated that universal topological properties can be measured from the non-equilibrium response of superconducting qubits. With these experiments as a guidepost, I discuss how these robust probes can be extended to larger many-body systems, where they may be a useful tool in probing novel quantum states and performing metrology.