Explore statistical shape analysis through the lens of persistent homology in this 51-minute lecture by Katharine Turner. Delve into the Persistent Homology Transform and its applications within the field of Applied and Computational Algebraic Topology. Learn about classical shape statistics, related variants, and TDA approaches. Examine approximation techniques, multidimensional scaling, and alignment methods. Investigate boundary considerations and motivation examples. Gain insights into recent progress, generic provisos, and the process of defining pairwise distances using Persistent Homology Transforms. This comprehensive talk, presented at the Hausdorff Center for Mathematics, offers a deep dive into cutting-edge techniques for analyzing and comparing shapes statistically.