Abstract: I will discuss three closely-related moduli problems: moduli of representations of star-shaped quivers, moduli of hyperpolygons, and moduli of parabolic Higgs bundles. One theme that weaves these three problems together is complete integrability. I will discuss recent results on the topology of these moduli spaces (joint work with Jonathan Fisher) and then pose questions on the relationship between stability for Higgs bundles and stability for hyperpolygons.