Explore the mathematical foundations of homogeneous Einstein manifolds in this 54-minute conference talk delivered at the International Centre for Theoretical Sciences. Delve into the geometric properties and theoretical implications of these special Riemannian manifolds where the Einstein tensor is proportional to the metric tensor, while maintaining homogeneity under group actions. Examine recent developments and revisit classical results in this intersection of differential geometry and mathematical physics. Learn about the classification problems, construction methods, and applications of homogeneous Einstein manifolds in both pure mathematics and theoretical physics contexts. Discover how these geometric structures contribute to our understanding of spacetime models and provide insights into the relationship between symmetry and curvature in Riemannian geometry.