  We extend Sklyanin's method of separation of variables to quantum integrable
models associated to elliptic curves. After reviewing the differential case,
the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider
the difference case and find a class of transfer matrices whose eigenvalue
problem can be solved by separation of variables. These transfer matrices are
associated to representations of the elliptic quantum group
$E_{\tau,\eta}(sl_2)$ by difference operators. One model of statistical
mechanics to which this method applies is the IRF model with antiperiodic
boundary conditions. The eigenvalues of the transfer matrix are given as
solutions of a system of quadratic equations in a space of higher order theta
functions.
