Persistent homology tracks the evolution of topological features (connected components H₀, loops H₁) as a filtration parameter ε increases — edges appear when two points come within distance ε. Features are born when they first appear and die when they merge into older features. The persistence diagram plots (birth, death) pairs: points far from the diagonal represent significant topological structure that persists over long parameter ranges. The stability theorem (Cohen-Steiner et al., 2007) guarantees that small perturbations in the data produce small changes in the diagram under the bottleneck distance, making persistent homology robust to noise.