Topological data analysis (TDA) applies methods of topology in data analysis and found many applications in data science in the recent decade that go well beyond machine learning. TDA builds upon the observation that data often possesses a certain intrinsic shape such as the shape of a point cloud, the shape of a signal or the shape of a geometric object. Persistent homology is probably the most prominent tool in TDA that gives us the means to describe and quantify topological properties of these shapes.In this paper, we give an overview of the basic concepts of persistent homology by interweaving intuitive explanations with the formal constructions of persistent homology. In order to illustrate the versatility of TDA and persistent homology we discuss three domains of applications, namely the analysis of signals and images, the analysis of geometric shapes and topological machine learning. With this paper we intend to contribute to the dissemination of TDA and illustrate their application in fields that received little recognition so far, like signal processing or CAD/CAM.
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