Abstract: Topological data analysis (TDA) provides a mathematical framework for extracting multiscale geometric and topological structure from complex data. Unlike conventional methods designed for Euclidean data, TDA naturally analyzes point clouds, networks, images, and other non-Euclidean objects through persistent homology, graph filtrations, and related topological invariants. This tutorial introduces the statistical foundations of TDA, including simplicial complexes, filtrations, persistent homology, persistence diagrams, and topological distances, together with practical computation using the standalone open-source MATLAB toolbox PH-STAT (https://github.com/laplcebeltrami/PH-STAT). Hands-on examples will cover structural and functional brain connectomics, dynamic brain networks, and other non-Euclidean datasets. The tutorial will also introduce recent advances in topological analysis of directed interactions and causal modeling, including Hodge-theoretic representations of feedback loops and cyclic information flow. Participants will gain both the theoretical background and practical tools needed to apply topological methods to modern data analysis problems.