I will give a synopsis of an article by Robert Ghrist (Bull. AMS, 45, no. 1, Jan 2008, pp. 61-75). From the abstract: "This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets-persistent homology-and a novel representation of this algebraic characterization-barcodes. We sketch an application of these techniques to the classification of natural images."