Persistent homology, the flagship method of topological data analysis, can be used to provide a quantitative summary of the shape of data.  One way to pass data to this method is to start with a finite, discrete metric space (whether or not it arises from a Euclidean embedding) and to study the resulting filtration of the Rips complex.  In this talk, we will discuss a new suite of methods for turning a time series into a network, and how to use persistent homology to quantify the structure of this data.  Combined with machine learning methods, we show how this can be used to classify behavior in time series in both synthetic and experimental data.