Explore the advanced mathematical concepts 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. Examine recent developments in the topological and geometric properties of quiver moduli spaces, and discover their applications in Donaldson-Thomas and Gromov-Witten theory. Learn about the motivation behind studying these spaces from a representation theory perspective, and understand their construction using Geometric Invariant Theory. Investigate various examples, analyze the topology and geometry of quiver moduli spaces, with a focus on their cohomology. Conclude by exploring the applications of quiver moduli spaces to Gromov-Witten and Donaldson-Thomas theory through the lens of wall-crossing.