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?



Classification of homogeneous Einstein manifolds
Yes, the question is still open.


I suggest to read [this quite recent paper](https://arxiv.org/abs/2107.06609) by Kerr and Böhm. It reviews some of the most important advances in the problem and includes several open problems.


