Long ago it was proved that the number of ends of a finitely presented infinite group is either 1, 2, or uncountably infinite. The semistability problem can be viewed as a refinement of this. Let G be a finitely presented group and let K be a finite complex whose fundamental group is G. The proper homotopy classes of proper rays in K can be regarded as “strong ends” of G. When G has one end, the number of strong ends is either 1 or is uncountably infinite. The semistability question is to decide whether this number is always 1. This problem is really a question about “the fundamental group at infinity” of G, and (via abelianization) to a structural question about H^2(G, ZG). When G is hyperbolic this number is always 1, but for CAT(0) groups the problem is still open. I will explain various recent developments on this problem in both the CAT(0} case and the general finitely presented case.