Recently, there has been a lot of work in developing algebraic and topological tools suitable for use on finite metric spaces (data), most notably persistent homology and level set techniques. These both involve functors from a real ordered indexing set to \(\mathrm{Top}\) to \(k\)-vector spaces, in which a classification theorem allows for a decomposition of indexed modules into intervals, under some conditions. These intervals, which commonly represent persistent homology classes, are then used to depict the results as barcodes or persistence diagrams, which can be analyzed statistically. With the right structure on the category of barcodes, this last transformation can also be defined as an equivalence of categories (Kashiwara-Schapira 2017, related work by Bauer-Lesnick 2016). I'll discuss the use and limits of this approach in TDA, and, time permitting, some applications of persistence techniques in topology and geometry. This is an expository talk.