Geometric class field theory (curves over a finite field) has been generalized to higher dimensional varieties over a finite field (and other arithmetical fields). Some of the key names here are Lang, Serre, Bloch, Kato-Saito...

Question: Does there exist a precise formulation of Langlands's conjectures which constitutes a non-abelian generalization of these results?

Probably relevant: Kazhdan's talk here at the Panel on Open Questions held at Gross's 60th Birthday Conference.