# Classification of homogeneous Einstein manifolds

In Besse's "Einstein manifolds", p. 177, he states that, until that moment, no general classification of homogeneous Einstein manifolds was know, even in the compact case. More specifically, he poses a problem: classify the compact simply connected homogeneous manifolds $$M=G/K$$ which admit a $$G$$-invariant Einstein metric.

Does that question remain open to this day?