This is joint with Megumi Harada of McMaster University. I will discuss our recent progress on the question of hyperkahler Kirwan surjectivity for quiver varieties. We develop a Morse theory for the norm-square of a moment map on the space of representations of a quiver, and show that the algebro-geometric Harder-Narasimhan stratification of this space coincides with the analytically-defined Morse-theoretic stratification associated to the norm-square of the moment map. This is the first step in a program to study hyperkahler quotients by developing a Morse theory for singular spaces. The work in this direction is still ongoing, however at this stage our approach provides some insight into the topology of specific examples of small-rank quiver varieties, including hyperpolygon spaces and some ADHM quivers. The first half of the talk will be expository and contain basic definitions and examples of quiver varieties.