Persistent homology is an algebraic tool for measuring topological features of shapes and functions. Given a filtration of a topological space (a nested sequence of subspaces), it determines the lifespan of homological features (connected components, tunnels, voids) within the filtration.The course will cover the fundamental ideas of persistent homology, fundamental results such as the stability of persistence diagrams, and motivating applications such as homology inference of shapes from point clouds and topological simplification of scalar functions.