  In this article, we consider Nakajima quiver varieties from the point of view
of symplectic algebraic geometry. We prove that they are all symplectic
singularities in the sense of Beauville and completely classify which admit
symplectic resolutions. Moreover we show that the smooth locus coincides with
the locus of canonically $\theta$-polystable points, generalizing a result of
Le Bruyn; we study their \'etale local structure and find their symplectic
leaves. An interesting consequence of our results is that not all symplectic
resolutions of quiver varieties appear to come from variation of GIT.
