  We review recent results on the finite-gap properties of difference operators
with elliptic coefficients and give explicit characterization of spectral
curves for difference analogues of the higher Lam\'e operators. This curve
parametrizes double-Bloch solutions to the difference Lam\'e equation. The
curve depends on a positive integer number \ell, related to its genus g by
g=2\ell, and two continuous parameters: the lattice spacing \eta and the
modular parameter \tau. Isospectral deformations of the difference Lam\'e
operator under Volterra flows are also discussed.
