A classical exact sequence which defines the ideal class group 
of a number field may be extended to arbitrary tori (the classical case
being that of the trivial torus G_m). This exact sequence for tori has
an analog for abelian varieties, which leads to the definition of the
Neron class group of an abelian variety A (this group is closely
connected to the groups of components of the Neron model of A). In this
talk I will show the connections that there exist between this class
group and the better-known Tate-Shafarevich group of A. I will discuss
open problems and the connections of these to the BSD conjecture (work
in progress).