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It has found a\r\nbroad range of applications, both within and outside mathematics,\r\nincluding to matroid theory, hyperplane arrangements, and data analysis.\r\n\r\nI will discuss two of my own contributions to discrete homotopy theory,\r\none more theoretical and one more applied. The first is a proof, joint\r\nwith D. Carranza (Compos. Math., 2024), of the conjecture by E. Babson,\r\nH. Barcelo, M. de Longueville, and R. Laubenbacher that discrete\r\nhomotopy groups can be topologically realized. The second, joint with N.\r\nKershaw (arXiv:2506.15020), builds on this result and introduces a new\r\nmethod of data analysis, which we call persistent discrete homology. We\r\nshow that in addition to its utility for clustering, it can detect other\r\ngeometric features of a data set. 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