The author considers the existence of Kähler-Einstein metrics on a natural generalization of toric Fano manifolds: the bi-equivariant Fano compactifications of complex connected reductive groups. More precisely, given such a group \(G\), he considers the Fano manifolds \(X\) which admit a \(G\times G\)-action with an open dense orbit isomorphic to \(G\) as a \(G\times G\)-homogeneous space under left and right translations. He calls these manifolds group compactifications. As in the toric case, a polytope \(P^+\) is associated to such a manifold, that encodes the information about the boundary \(X\backslash G\) and the anticanonical line bundle. The main theorem is the following: A Fano compactification of \(G\) admits a Kähler-Einstein metric if and only if \(\text{bar}_{DH}(P^+)\in 2\rho+\Xi\). Here \(\text{bar}_{DH}(P^+)\) is the barycenter of \(P^+\) and \(\Xi\) the relative interior of the cone generated by \(\Phi^+\), a root system of \(G\) (\(\rho\) is half the sum of positive roots).