For context for Tom's answer,
let me state the naive version of the conjecture, which has been around since around 1997 I think (due to Beilinson-Drinfeld). It calls for an equivalence of (dg) categories
$$D(Bun_G(X))\simeq QC(Loc_{G^\vee}(X))$$
between (quasi)coherent $D$-modules on the stack of $G$-bundles on a curve $X$, and (quasi)coherent sheaves on the (derived) stack of flat $G^\vee$ connections on the curve.
Moreover (and this is where most of the content lies) this equivalence should be an equivalence as module categories for the spherical Hecke category $Rep(G^\vee)$ acting on both sides for every choice of point $x\in X$. The action on the right is given by simple multiplication operators (tensor product with a tautological vector bundle on $Loc_{G^\vee}(X)$ attached to a given representation of $G^\vee$ and point $x$. On the right hand side the action is by convolution (Hecke) functors, associated to modifications of $G$-bundles at $x$ (relative positions at $x$ of $G$-bundles are labeled by the affine Grassmannian, and in order to formulate this statement we use the geometric Satake theorem of Lusztig, Drinfeld, Ginzburg and Mirkovic-Vilonen).
The equivalence can be further fixed uniquely by using "Whittaker normalization",
a geometric analog of the identification of L-functions in the classical Langlands story.
In this form the conjecture is a theorem for $GL_1$, using the extension of the Fourier-Mukai transform for D-modules. There are no other groups for which it's known (yet - though perhaps Dennis already has $SL_2$), and few curves (one can do $P^1$ for example). Also (as Scott points out) this is only the unramified case, and there are natural conjectures to make with at least tame ramification (a "parabolic" version of the above).
There's also a close variant of this conjecture which comes naturally from S-duality for N=4 super-Yang-Mills, thanks to Kapustin-Witten. [Edit: in fact the physics suggests a far more refined version of the whole geometric Langlands program.]
It is a theorem of Beilinson-Drinfeld if we restrict on the left hand side to D-modules generated by D itself, and on the right to coherent sheaves living on opers.
It is also well known that this conjecture as stated is too naive, due to bad "functional analysis" of the categories involved (precisely analogous to the analytic issues appearing eg in the Arthur-Selberg trace formula). On the D-module side, the stack $Bun_G$ is not of finite type, and one might want to make more precise "growth conditions" on D-modules along the Harder-Narasimhan strata. On the coherent side, $Loc$ is a derived stack and singular, and one ought to modify the sheaves allowed at the singular points -- in particular at the most singular point, the trivial local system. [Edit:One approach to correcting this involves the "Arthur $SL_2$" -- roughly speaking looking at local systems with an additional flat $SL_2$-action that controls the reducibility..this comes up really beautifully from the physics in work of Gaiotto-Witten, one of the first points where the physics is clearly "smarter" than the math.] These two issues are very neatly paired by the duality. If one restricts to irreducible local systems
and "cuspidal" D-modules, these issues are completely avoided (though the conjecture even on this locus remains open except for $GL_n$, where I believe it can be deduced from work of Gaitsgory following his joint work with Frenkel-Vilonen, see his ICM).
In any case, Dennis has now given a precise formulation of a conjecture, which was at some point at least on his website and follows the outline Tom explained. It is very clear that homotopical algebra a la Lurie is crucial to any attempt to prove this, and Dennis and Jacob have made (AFAIK) great progress on this. The basic idea of the approach is the same as that carried out by Beilinson-Drinfeld and suggested by (old) conformal field theory (in particular Feigin-Frenkel) and (new) topological field theory (Kapustin-Witten and Lurie) -- i.e. a local-to-global argument, deducing the result from a local equivalence which comes from representation theory of loop algebras. The necessary machinery for "categorical harmonic analysis" is now available, and I'm looking forward to hearing a solution before very long..
Edit: In response to Kevin's comment I wanted to make some very informal remarks about ramification.
First of all one needs to keep in mind the distinction between reciprocity (which is what the above conjecture captures) and functoriality. In the geometric setting the former is strictly stronger than the latter, while in the arithmetic setting most of the emphasis is on the latter. It is in fact quite easy to formulate a functoriality conjecture in the geometric setting with arbitrary ramification. Namely, fix some ramification and look at the category of D-modules on the stack of G-bundles with corresponding level structure. Then this is a module category over coherent sheaves on the stack of $G^\vee$ connection
with poles prescribed by the ramification (eg we can take full level structure and allow arbitrary poles). Then one can conjecture that given a map of L-groups $G^\vee\to H^\vee$ the corresponding automorphic categories are given simply by tensoring the module categories from $QC(Loc_{G^\vee}(X))$ to $QC(Loc_{H^\vee}(X))$ (everything here must be taken on the derived level to make sense). It is not hard to see this follows from any form of reciprocity you can formulate. And there is also a geometric version of the Arthur-Selberg trace formula in the ramified setting (under development).
Second, one can make a reciprocity conjecture with full ramification, though you have todecide to what extent you believe it. In the "completed"/"analytic" form of geometric Langlands that comes out of physics such a conjecture in fact appears in a paper of Witten on Wild Ramification. Roughly speaking in the above reciprocity rather than just looking
at the module category structure for D-modules over QC of local systems, you can ask for them to be equivalent... not stated anywhere since maybe I'm too naive and this is known to be too far from the truth, but I think more likely people haven't thought about it very much.
Third, the local story: of course in the p-adic case Kevin discusses one restricts to $GL_n$. There are two things to point out: first of all in the geometric setting one is interested in all groups, and $GL_n$ is not much simpler as far as our understanding of the local story goes. Second, while everything is much harder and deeper in the p-adic setting than the geometric setting, it's worth pointing out that a formulation of a local geometric Langlands conjecture is a much more subtle proposition, involving the representation theory of loop groups on derived categories which is only beginning to be within the reach of modern technology (even at the level of formulation of the objects!)
That being said, there are rough forms of local geometric Langlands conjectures developed
by Frenkel, Gaitsgory and Lurie. It is best understood in the so-called "quantum geometric Langlands program", a deformation of the above picture involving the representation theory of quantum groups, where Gaitsgory-Lurie give a precise general local conjecture and make progress on its resolution. The usual case above is a bit degenerate and one needs to be more careful. In any case the rough form of the local conjecture is an equivalence of 2-categories (again everything has to be taken in the appropriate derived sense) between "smooth" LG-actions on categories and quasicoherent sheaves of categories over the stack of connections on the punctured disc..
--THAT being said we don't have a proof of this for $GL_n$ so again the number theorists win! just wanted to give some sense that there is a reasonable understanding of full ramification.
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.
Best Answer
OK, this is a very broad question so I'll be telegraphic. There is a sequence of increasingly detailed conjectures going by the name GL -- it's really a "program" (harmonic analysis of $\mathcal{D}$-modules on moduli of bundles) rather than a conjecture -- and only the first of this sequence has been proved (and only for $GL_n$), but I don't want to get into this.
There are several kinds of reasons you might want to study geometric Langlands:
direct consequences. One application is Gaitsgory's proof of de Jong's conjecture (arXiv:math/0402184). If you prove the ramified geometric Langlands for $GL_n$, you will recover L. Lafforgue's results (Langlands for function fields), which have lots of consequences (enumerated eg I think in his Fields medal description), which I won't enumerate. (well really you'd need to prove them "well" to get the motivic consequences..) In fact you'll recover much more (like independence of $l$ results). To me though this is the least convincing motivation..
Original motivation: by understanding the function field version of Langlands you can hope to learn a lot about the Langlands program, working in a much easier setting where you have a chance to go much further. In particular the GLP (the version over $\mathbb{C}$) has a LOT more structure than the Langlands program -- ie things are MUCH nicer, there are much stronger and cleaner results you can hope to prove, and hope to use this to gain insight into underlying patterns.
By far the greatest example of this is Ngo's proof of the Fundamental Lemma --- he doesn't use GLP per se, but rather the geometry of the Hitchin system, which is one of the key geometric ingredients discovered through the GLP. To me this already makes the whole endeavor worthwhile..
a. The first is the theory of integrable systems -- many classical integrable systems fit into the Hitchin system framework, and geometric Langlands gives you a very powerful tool to study the corresponding quantum integrable systems. In fact you (namely BD) can motivate the entire GLP as a way to fully solve a collection of quantum integrable systems. This has has lots of applications in the subject (eg see Frenkel's reviews on the Gaudin system, papers on Calogero-Moser systems etc).
b. The second is conformal field theory (again BD) --- they develop CFT (conformal, not class, field theory!) very far towards the goal of understanding GLP, leading to deep insights in both directions (and a strategy now by Gaitsgory-Lurie to solve the strongest form of GLP).
c. The third is four-dimensional gauge theory (KW). To me the best way to motivate geometric Langlands is as an aspect of electric-magnetic duality in 4d SUSY gauge theory. This ties in GLP to many of the hottest current topics in string theory/gauge theory (including Dijkgraaf-Vafa theory, wall crossing/Donaldson-Thomas theory, study of M5 branes, yadda yadda yadda)...
There's more but this is already turning into a blog post so I should stop.