  We study asymptotics of the dimer model on large toric graphs. Let $\mathbb
L$ be a weighted $\mathbb{Z}^2$-periodic planar graph, and let $\mathbb{Z}^2 E$
be a large-index sublattice of $\mathbb{Z}^2$. For $\mathbb L$ bipartite we
show that the dimer partition function on the quotient
$\mathbb{L}/(\mathbb{Z}^2 E)$ has the asymptotic expansion $\exp[A f_0 +
\text{fsc} + o(1)]$, where $A$ is the area of $\mathbb{L}/(\mathbb{Z}^2 E)$,
$f_0$ is the free energy density in the bulk, and $\text{fsc}$ is a finite-size
correction term depending only on the conformal shape of the domain together
with some parity-type information. Assuming a conjectural condition on the zero
locus of the dimer characteristic polynomial, we show that an analogous
expansion holds for $\mathbb{L}$ non-bipartite. The functional form of the
finite-size correction differs between the two classes, but is universal within
each class. Our calculations yield new information concerning the distribution
of the number of loops winding around the torus in the associated double-dimer
models.
