**Abstract:** A stratification of a topological space (with singularities) is a decomposition of the space into manifold pieces, of various dimensions, that all "fit together nicely" in some precisely-defined sense. In this talk, we'll discuss ways in which algebraic theories about stratified spaces can be combined with persistent homology methods to give insight about actual data. First, we imagine a situation where point cloud data is drawn from or near a stratified space that is embedded in Euclidean space, and our goal is to gain information about the stratification from the point cloud. We give conditions under which this can be done; the main tools are a multi-scale notion of persistent local homology, as well as (co)kernel persistent homology, and the end result is a family of clusterings of the point cloud into groupings which could be placed into the same stratum at different radius scales. Next, we take a different perspective, imagining that we are given a topological space with a fixed stratification endowed with a height function. We show how intersection homology, a homology theory that is particularly sensitive to singularities, can be incorporated into persistent homology theory, giving information about how the height function filters the space and its singularities. Finally, we describe some very recent initial steps towards combining these two theories, illustrating how zig-zag persistent homology can be used to measure changes in the intersection homology of a stratified space, as the stratification itself changes in a continuous fashion.