Difference and discrete equations on the real axis and the semiaxis

We consider general difference equations on the real axis and the semiaxis in terms of the theory of pseudo-differential equations and boundary value problems. It is shown that a principal role for describing the picture of solvability for such equations plays an index of factorization of an elliptic symbol of difference equation with constant coefficients. A form of solution of such equation in the spaces of square integrable functions is described and necessary and sufficient conditions for solvability are obtained. Some discrete analogues of such equations are described.