This article surveys the work of Carlsson and collaborators on applying computational algebraic topology to feature detection and shape recognition in high-dimensional data. The primary mathematical tool discussed is persistent homology, a homology theory for point-cloud data sets, and its novel representation, barcodes. The author introduces the concept of converting point clouds into global topological objects through simplicial complexes, specifically the Čech and Rips complexes, and explains how these complexes can be used to infer global structure from discrete data. The article also delves into the theory of persistent homology, which allows for the identification of essential topological features by distinguishing between persistent and transient features over a range of parameters. Barcodes, a graphical representation of persistent homology, are introduced as a way to visualize and interpret these features. The article concludes with an example of applying these techniques to natural images, demonstrating how persistent homology can uncover meaningful structures and provide insights into the underlying data.This article surveys the work of Carlsson and collaborators on applying computational algebraic topology to feature detection and shape recognition in high-dimensional data. The primary mathematical tool discussed is persistent homology, a homology theory for point-cloud data sets, and its novel representation, barcodes. The author introduces the concept of converting point clouds into global topological objects through simplicial complexes, specifically the Čech and Rips complexes, and explains how these complexes can be used to infer global structure from discrete data. The article also delves into the theory of persistent homology, which allows for the identification of essential topological features by distinguishing between persistent and transient features over a range of parameters. Barcodes, a graphical representation of persistent homology, are introduced as a way to visualize and interpret these features. The article concludes with an example of applying these techniques to natural images, demonstrating how persistent homology can uncover meaningful structures and provide insights into the underlying data.