Abstract: Topological Data Analysis provides a framework for analyzing data that is robust to perturbations of the data. One way in which it accomplishes this is by introducing stable invariants: invariants of data sets that vary continuously with respect to suitable metrics on the collection of data sets. In this talk I will present several well known invariants of different kinds of data (such as metric spaces, metric measure spaces, dynamic metric spaces, and filtered metric spaces) and a theoretical framework that lets us prove known stability results as well as novel ones.