Explore discrete homotopy theory in this 57-minute lecture by Chris Kapulkin from the Applied Algebraic Topology Network. Discover this mathematical field that applies continuous homotopy techniques to simple graphs, introduced about 20 years ago by Barcelo and collaborators building on Atkin's work from the mid-1970s. Learn about its diverse applications in matroid theory, hyperplane arrangements, topological data analysis, time series analysis, and even understanding how social interactions between preschoolers affect academic performance. The lecture highlights recent theoretical and computational advances, including the resolution of the 2006 Babson-Barcelo-de Longueville-Laubenbacher conjecture (joint work with Carranza, published in Compositio Mathematica, 2024) and an efficient algorithm for computing discrete homology groups (joint work with Kershaw, arXiv:2410.09939).