We will discuss Einstein manifolds which are invariant under an isometric Lie group action. Our main goal is to explain the proof of the 1975 Alekseevskii Conjecture on non-compact homogeneous Einstein spaces, recently obtained in collaboration with Christoph Böhm (Münster). To that end, we will also present new structure results for Einstein metrics on principal bundles. The talk will conclude with open questions and future directions.