A Riemannian manifold is said to be Einstein if it has constant Ricci curvature, i.e., if its metric \(g\) satisfies Ric\(_g=cg\). When working in a homogeneous space, this condition turns into a collection of algebraic equations. Despite this apparent simplicity, the study of homogeneous Einstein manifolds turns out to be very involved and is, up to this day, a vibrant field of study in differential geometry. In this expository paper, the author looks at the fundamental properties and techniques in the study of homogeneous Einstein metrics, and the progress made in recent years on the topic. The cases of compact and noncompact homogeneous Einstein spaces are treated separately: In the compact case, a given homogeneous space \(G/K\) may admit several \(G\)-invariant Einstein metrics. In contrast, a homogeneous Einstein space of negative scalar curvature is isometric to a solvable Lie group with a left-invariant metric, and a given solvable Lie group admits a unique Einstein metric up to scaling and isometry. The survey also lists some open problems and provides the reader with an extensive list of references for those who want to dig deeper into the topic.