EDIT Few days ago a survey by A. Parshin appeared in arxiv.
I think it is the best place to look on the higher-dimensional Langlands.
From the abstract:
A brief survey is given of the classical Langlands correspondence
between n-dimensional representations of Galois groups of local and
global fields of dimension 1 and irreducible representations of the
groups GL(n). A generalization of the Langlands program to fields of
dimension 2 is considered and the corresponding version for
1-dimensional representations is described. We formulate a conjecture
on a direct image (=automorphic induction) of automorphic forms which
links the Langlands correspondences in dimension 2 and 1. The direct
image conjecture implies the classical Hasse-Weil conjecture on the
analytical behaviour of the L-functions of curves defined over global
fields of dimension 1.
Actually similar question has been asked. Higher-dimensional Langlands is something very intriguing. I did not follow recent advances, but let me mention some part which I know about.
Let us start with NON-geometric local case. Langlands correspondence is roughly speaking "bijection" between representations of Galois group and representations of GL(Local Field).
Abelian case (class field theory) is bijection between characters of Galois and characters Local field, if dualize we get Galois/[Galois,Galois] = (Local Field)^*.
One of the first questions to ask - whether it is possible to generalize local class field theory to higher dimensions ?
The main idea by A. Parshin is that in n-dimensions one should consider Milnor's (n)-th K-group of local field instead of (Local Field)^. In particular for n=1 K_1^Milnor(Field)=Field^. (By the way the definition of higher dimensional local field should also be given).
Parshin also found higher analogs of various symbols and proved higher analogs of reciprocity laws.
To the best of my knowledge there were no further developments in the field before Kapranov's paper "Analogies between the Langlands correspondence and topological quantum field theory" . In that paper he gave certain vision what higher dimensional Langlands might be.
His idea (quite amazing) that "representations" should be substituted by "k-representations" (i.e. representations in higher categories), (e.g. for surfaces we should consider 2-representations).
So in n-th dimensions k-representations of dimension r of Galois group should correspond to (n-k)-representations of GL_r(n-Local Field)
In particular abelian version will correspond to Parshin's higher dimensional class field theory, since Milnor's K-groups correspond higher-representations.
Now about the geometric version. It is quite unclear for me.
In 1-dimension, we can think of flat connections as analogs of Galois representations.
It seems in higher dimensions we should consider gerbes with flat connections
as analogs of higher representations of Galois group, whatever it means...
The geometric substitute for moduli space of vector bundles and Hecke-eigen sheaves is not clear for me. The reason is that in 1-dimension moduli space of vector bundles arise as standard coset G_{out}\G(k((z))/G_{in}, however in higher dimensions I do not see how the group G( k((z,u)) ) may have finite-dimensional quotient.
So I do not see something finite-dimensional where "n-Hecke eigensheaf" may live.
Well, it is probably just my own problem. I have speculated around these things in http://arxiv.org/abs/hep-th/0604128, but I am afraid it is very unclear...
Last year there appeared a paper Unramified two-dimensional Langlands correspondence,
which probably is the last mile achievement in the question. Unfortunately I have no time to follow these developments.
