Separation of variables for quantum integrable systems on elliptic curves

We extend Sklyanin's method of separation of variables to quantum integrabl e models associated to elliptic curves. After reviewing the differential ca se, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we c onsider the difference case and find a class of transfer matrices whose eig envalue problem can be solved by separation of variables. These transfer ma trices are associated to representations of the elliptic quantum group E-ta u,E-eta(sl(2)) by difference operators. One model of statistical mechanics to which this method applies is the interaction-round-a-Face model with ant iperiodic boundary conditions. The eigenvalues of the transfer matrix are g iven as solutions of a system of quadratic equations in a space of higher-o rder theta functions.