In the late 1980s Tian proved the existence of Kaehler-Einstein metrics on all Fano surfaces for which the question had been open, using the "alpha- invariants" \alpha_{m,1}(M) and, in certain delicate cases, \alpha_{m,2}(M).

I will discuss a conjectural approach to proving the existence of Kaehler-Einstein metrics on Fano manifolds in similarly delicate cases in higher dimension, using the higher alpha-invariants \alpha_{m,k}(M).