After reviewing the classical theory of quiver moduli spaces via Mumford’s reductive GIT, I will report on joint work with Victoria Hoskins and Tanguy Vernet which constructs moduli spaces for representations of quivers with multiplicities. Along the way I will indicate the challenges posed by this generalisation, namely the non-reductivity of the groups we need to quotient by, and outline the theoretical developments (joint with Eloise Hamilton and Victoria Hoskins) that make this possible.