  We study chip-firing on a signed graph $G_\phi$, employing a general theory
of chip-firing on invertible matrices introduced by Guzm\'an and Klivans. Here
a negative edge designates an adversarial relationship, so that firing a vertex
incident to such an edge leads to a loss of chips at both endpoints. The
chip-firing rule for $G_\phi$ is described by its reduced Laplacian matrix
$L_{G_\phi}$, which also defines the critical group ${\mathcal K}(G_\phi)$. The
valid chip configurations are given by the lattice points of a rational cone
determined by $G_\phi$ and the underlying graph $G$. This gives rise to notions
of critical as well as $z$-superstable configurations, both of which are
counted by the determinant of $L_{G_\phi}$. We establish general results
regarding these configurations, focusing on efficient methods of verifying the
underlying properties. We then study the critical groups of signed graphs in
the context of vertex switching and Smith normal forms. We use this to compute
the critical groups of various classes of signed graphs including signed
cycles, wheels, complete graphs, and fans, in the process generalizing results
of Biggs and others.
