A key development of modern condensed matter theory has been the discovery of topological phases of quantum matter that are stabilised by underlying symmetries. A central role is played by time-reversal symmetry (TRS), which leads to various topological phases, notably topological band insulators and the Haldane phase of quantum magnets. In this talk I shall discuss the fate of topological phases in situations that go beyond ground state physics: in non-equilibrium situations, and in settings where the system is coupled to an external bath. I will show that, while topological features can be preserved in these situations, TRS-protected phenomena are intrinsically fragile. These results highlight the enigmatic nature of TRS in quantum mechanics and elucidate potential challenges in utilizing topological systems for quantum technologies.