  We study algebro-geometric (finite-gap) and elliptic solutions of fully
discretized KP or 2D Toda equations. In bilinear form they are Hirota's
difference equation for $\tau$-functions. Starting from a given algebraic
curve, we express the $\tau$-function and the Baker-Akhiezer function in terms
of the Riemann theta function. We show that the elliptic solutions, when the
$\tau$-function is an elliptic polynomial, form a subclass of the general
algebro-geometric solutions. We construct the algebraic curves of the elliptic
solutions. The evolution of zeros of the elliptic solutions is governed by the
discrete time generalization of the Ruijsenaars-Schneider many body system. The
zeros obey equations which have the form of nested Bethe-Ansatz equations,
known from integrable quantum field theories. We discuss the Lax representation
and the action-angle-type variables for the many body system. We also discuss
elliptic solutions to discrete analogues of KdV, sine-Gordon and 1D Toda
equations and describe the loci of the zeros.
