This article surveys recent work by Carlsson and collaborators on applying computational algebraic topology to feature detection and shape recognition in high-dimensional data. The primary mathematical tool is persistent homology, a homology theory for point-cloud data sets, and its novel representation—barcodes. The article sketches an application of these techniques to the classification of natural images. Data is often represented as an unordered sequence of points in Euclidean space. Data from sensor readings, questionnaires, or ecosystems often resides in high-dimensional space. The global shape of data can provide important information about the underlying phenomena. Point cloud data from physical objects in 3D can be used to represent objects, and motion-capture data can be used to record geometric points over time. The goal is to identify and recognize global features in such data. Topological data analysis involves inferring high-dimensional structure from low-dimensional representations and assembling discrete points into global structure. The article discusses converting data points into global topological objects using simplicial complexes indexed by a proximity parameter. These complexes are analyzed using algebraic topology, specifically persistent homology, which is encoded in a parameterized version of a Betti number—barcodes. Persistent homology is a rigorous method for identifying topological features that persist over a significant parameter range. It is used to analyze sequences of Rips or Čech complexes. The persistent homology of a data set is represented as a barcode, a graphical representation of homology groups as intervals. Barcodes are stable in the presence of noise and can reveal significant features in data. The article also discusses the computation of persistent homology and its application to natural images. A data set of 4,167 digital photographs was analyzed, revealing topological structures such as loops and circles. Persistent homology was used to identify these structures, showing that the data set is diffused around a primary circle in the 7-sphere. Further analysis revealed secondary circles, which correspond to different patterns in the data. The persistent homology of the data set also suggested a two-dimensional completion of the low-k persistent H₁ basis into a Klein bottle. The article concludes that persistent homology provides a powerful tool for analyzing high-dimensional data, revealing meaningful structures that are not discernible through other methods. It emphasizes the importance of topological data analysis in understanding complex data sets and its potential applications in various fields.This article surveys recent work by Carlsson and collaborators on applying computational algebraic topology to feature detection and shape recognition in high-dimensional data. The primary mathematical tool is persistent homology, a homology theory for point-cloud data sets, and its novel representation—barcodes. The article sketches an application of these techniques to the classification of natural images. Data is often represented as an unordered sequence of points in Euclidean space. Data from sensor readings, questionnaires, or ecosystems often resides in high-dimensional space. The global shape of data can provide important information about the underlying phenomena. Point cloud data from physical objects in 3D can be used to represent objects, and motion-capture data can be used to record geometric points over time. The goal is to identify and recognize global features in such data. Topological data analysis involves inferring high-dimensional structure from low-dimensional representations and assembling discrete points into global structure. The article discusses converting data points into global topological objects using simplicial complexes indexed by a proximity parameter. These complexes are analyzed using algebraic topology, specifically persistent homology, which is encoded in a parameterized version of a Betti number—barcodes. Persistent homology is a rigorous method for identifying topological features that persist over a significant parameter range. It is used to analyze sequences of Rips or Čech complexes. The persistent homology of a data set is represented as a barcode, a graphical representation of homology groups as intervals. Barcodes are stable in the presence of noise and can reveal significant features in data. The article also discusses the computation of persistent homology and its application to natural images. A data set of 4,167 digital photographs was analyzed, revealing topological structures such as loops and circles. Persistent homology was used to identify these structures, showing that the data set is diffused around a primary circle in the 7-sphere. Further analysis revealed secondary circles, which correspond to different patterns in the data. The persistent homology of the data set also suggested a two-dimensional completion of the low-k persistent H₁ basis into a Klein bottle. The article concludes that persistent homology provides a powerful tool for analyzing high-dimensional data, revealing meaningful structures that are not discernible through other methods. It emphasizes the importance of topological data analysis in understanding complex data sets and its potential applications in various fields.