  A new approach to the analytic theory of difference equations with rational
and elliptic coefficients is proposed. It is based on the construction of
canonical meromorphic solutions which are analytical along "thick paths". The
concept of such solutions leads to a notion of local monodromies of difference
equations. It is shown that in the continuous limit they converge to the
monodromy matrices of differential equations. New type of isomonodromic
deformations of difference equations with elliptic coefficients changing the
periods of elliptic curves is constructed.
