  A special case of the geometric Langlands correspondence is given by the
relationship between solutions of the Bethe ansatz equations for the Gaudin
model and opers - connections on the projective line with extra structure. In
this paper, we describe a deformation of this correspondence for $SL(N)$. We
introduce a difference equation version of opers called $q$-opers and prove a
$q$-Langlands correspondence between nondegenerate solutions of the Bethe
ansatz equations for the XXZ model and nondegenerate twisted $q$-opers with
regular singularities on the projective line. We show that the
quantum/classical duality between the XXZ spin chain and the trigonometric
Ruijsenaars-Schneider model may be viewed as a special case of the
$q$-Langlands correspondence. We also describe an application of $q$-opers to
the equivariant quantum $K$-theory of the cotangent bundles to partial flag
varieties.
