Abstract: [This is a tutorial style 2 hour talk] Topological Data Analysis (TDA) is now around 15 years old, and it is becoming a widely-used technique in data analysis, often in combination with more traditional methods. One of the key tools in TDA is the persistence diagram, which provides a compact description of the multi-scale topological information carried by a point cloud or other embedded object. In this informal tutorial, I will start by going over the very basics of persistent homology, showing how one can go from a point cloud, or a space equipped with a function, and produce a persistence diagram. Depending on audience interest, I can then cover some or all of the following: -some simple (and not-so-simple) applications which show how persistence diagrams, in combination with statistical ideas, can extract interesting information from unusual datasets. -the geometry of the space of persistence diagrams: why it's strange, and the possible implications for statistics with diagrams. -different algorithms for computing persistence diagrams, some of which are correct and slow, some of which are correct (in certain circumstances) and fast, some of which are fast and approximate with good guarantees. -a demonstration of our new software package RCA, which gives a very fast (and correct!) computation of persistence diagrams for components and loops.