Generalized ideal class groups can be described adelically in terms of a coset space for the group GL1, and this in turn leads to a notion of "class number" (as the size of a certain set, if finite) for an arbitrary affine algebraic group over a global field. Related to this is the notion of the "Tate-Shafarevich set" of an algebraic group, which is tied up with questions relating global and local information. Finiteness of class numbers and Tate-Shafarevich sets for affine algebraic groups was proved by Borel and his coworkers over number fields, andif one grants the finiteness of Tate-Shafarevich groups for abelian varieties then Mazur showed how to get such finiteness for all algebraic group varieties over number fields (which has applications to the local-to-global principle for projective varieties over number fields).

The above methods do not apply over global function fields. After reviewing some history, I will explain the content of a recent classification theorem of "pseudo-reductive groups" proved jointly with Gabber and G. Prasad that makes it possible to prove the analogous finiteness theorems in the function field case away from characteristic 2. If time permits I will say something about how this classification theorem is used to get such results.