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, find their symplectic leaves,
and we describe the Namikawa Weyl group. An interesting consequence
of our results is that not all symplectic resolutions of quiver
varieties appear to come from variation of GIT.

We apply this to the $G$-character variety of a
compact Riemann surface of genus $g > 0$, when $G$ is
$\mathrm{SL}(n,\mathbb{C})$ or $\mathrm{GL}(n,\mathbb{C})$. We show
that these varieties are symplectic singularities and classify when
they admit symplectic resolutions: they do when $g = 1$ or
$(g,n)=(2,2)$ (assuming $n \geq 2$). This is analogous to the case
of a quiver with one vertex, $g$ arrows, and dimension vector $(n)$.