Explore the fascinating world of quiver moduli spaces in this comprehensive lecture from the Felix Klein Lectures 2020 series. Delve into the algebraic varieties that encode continuous parameters of linear algebra type classification problems. Discover how recent years have seen advancements in understanding their topological and geometric properties, with applications extending to Donaldson-Thomas and Gromov-Witten theory. Begin by examining the motivation behind studying quiver moduli spaces from a representation theory perspective, review their construction using Geometric Invariant Theory, and analyze various examples. Progress to an exploration of the topology and geometry of these moduli spaces, with a particular focus on their cohomology. Conclude by investigating the applications of quiver moduli spaces to Gromov-Witten and Donaldson-Thomas theory through the lens of wall-crossing.