This tutorial accompanies the Summer School on Data Science for Dynamical Systems, http://www.lorentzcenter.nl/lc/web/2019/1112/info.php3?wsid=1112&venue=Oort, at the Lorentz Center in Leiden, The Netherlands. It was written by Henry Adams, Melissa McGuirl, and Yitzchak Solomon, with the tutorial lead by Henry Adams and Pepijn Roos Hoefgeest. Please see the corresponding wiki page: https://github.com/henryadams/Leiden-PersistentHomology/wiki.