Let Q be a quiver with dimension vector d. The Harder-Narasimhan filtration on the category Rep(Q) gives an algebraic stratification of the vector space Rep(Q,d) of representations of fixed dimension. When the ground field is C, this stratification agrees with the Morse stratification coming from the norm-square of a moment map. I will give an explicit description of this stratification, and show that it leads to an inductive procedure for computing the cohomology ring of the moduli space of stable representations of Q. I will give a few worked examples, and time-permitting, I will describe some applications to Nakajima quiver varieties.