Abstract: Topology has emerged as an essential tool for classifying materials.  What makes topology special is its robustness to local deformations.  This robustness leads to extremely long-lived topological defects in magnetic or electronic systems and to potentially long-lived quantum states that may serve as quantum memories or the elements of a quantum computer.  I will discuss recent theoretical ideas (and related experiments) that allow us to probe or exploit this topological robustness, even far from equilibrium.  In some cases, topology can even constrain dynamical observables for a complex quantum system coupled to an infinite bath, making these (apparently complex) problems much simpler.