Abstract: The aim of this talk and the talks by S. Meinhardt in June is to sketch the proof of a recent result computing the intersection cohomology of moduli spaces of semistable quiver representations in terms of the corresponding motivic Donaldson-Thomas invariants. This first talk serves as an introduction to the proof, focussing on the quiver-theoretic methods, like the geometry of the Hilbert-Chow map from framed to unframed moduli, the concept of virtually small maps, and the overall strategy using the BBDG Decomposition Theorem.