Most of the traditional time-series analysis techniques that are used to study trajectories from nonlinear dynamical systems involve state-space reconstructions and clever approximations of asymptotic quantities, all in the context of finite and often noisy data. Few of these techniques work well in the face of nonstationarity. Embedding a time series that samples different dynamical systems at different times, for instance---and then calculating a long-term Lyapunov exponent---does not make sense. Computational topology offers some important advantages in situations like this. Topological descriptions of structure are inherently more qualitative, and thus more robust, than more-rigid geometrical characterizations. Even when the data contain finite amounts of noise, computational topology can produce provable results [Day et al. 2008, Mischaikow et al. 1999], and it is naturally immune to changes of scale and orientation that skew the data. Faced with a time series that samples a number of different dynamical systems, one can detect regime changes by looking for shifts in the topological structure of the reconstructed dynamics: e.g., nearby points whose immediate future paths are significantly different. This segmentation strategy works in some situations where others do not---if the different components of the signal overlap in state space, for instance. Once a signal has been segmented into components, one can compute topological "signatures" of those components---e.g., with witness complexes and Conley index theory. The notion of persistence can be leveraged to make appropriate choices for the different free parameters in these algorithms. It may also be possible to take advantage of persistence in order to handle the additional challenges that arise when the data arrive in a stream and must be analyzed 'on the fly.' In this situation, one does not have the luxury of post facto analysis of the full data set. Rather, one must detect shifts immediately--and then immediately start building up a new model from the incoming stream. This is joint work with Jim Meiss, Vanessa Robins, and Zach Alexander.