Exotic many-body phenomena are usually associated with the ground state of a time-independent Hamiltonian. It is natural to ask whether these physics can survive in a dynamic setting. Under a generic drive, the steady equilibrium state is most likely an infinite-temperature featureless thermal state. However, there exist exceptional cases where thermalization either does not happen or is delayed for a sufficiently long time, called non-equilibrium many-body systems. In this thesis, we study mechanisms that can generate non-equilibrium dynamics: many-body localization, prethermalization, and projective measurements. We then demonstrate that the resulting quantum states can host a wide variety of many-body phenomena similar to the ground state, focusing on three aspects: topology, localization, and spontaneous symmetry breaking.