Yau and Aubin showed that a compact Kahler manifold with ample canonical bundle admits a Kahler-Einstein metric. The question of whether there exist constant scalar curvature metrics in classes away from the canonical class remains open, and is expected to be related to a notion of stability in the sense of geometric invariant theory. This idea comes from a well-known conjecture of Yau. It will be discussed how the J-flow of Donaldson and X. X. Chen is connected to this problem via the functional known as the Mabuchi energy. We find necessary and sufficient conditions for convergence of the J-flow. And, when the J-flow develops singularities, we show that, in some cases, estimates can be derived away from a subvariety. These can be used to prove, in two dimensions, a weak form of a conjectural remark of Donaldson that if the J-flow does not converge then it should blow up over some curves of negative self-intersection. It will be discussed how these results can be applied to prove properness of the Mabuchi energy for some Kahler classes, and, conjecturally, how they relate to notions of stability due to Tian and Ross-Thomas. (Joint work with J. Song)