It is by now well-understood that gapped ground states of local Hamiltonians can be classified topologically, and the nontrivial states exhibit a variety of interesting topological phenomena. In this talk I’ll discuss how the topological classification can be generalized to disordered ensembles and mixed states. I’ll focus on short-range-entangled ensembles with global symmetries, with an important distinction between exact and average symmetries, the latter of which are only preserved by the ensemble as a whole but not on individual states. I’ll present a general framework based on domain wall decorations to classify topological phases in such ensembles, and discuss examples of “intrinsically disordered” average-symmetry-protected topological phases, which can not be realized in clean systems.