Abstract: Let $G$ be a compact connected Lie group and $K$ a closed connected subgroup. In 1950 Cartan determined the real cohomology ring of the homogeneous space $G/K$. The formula can be written as a torsion product involving the cohomologies of the classifying spaces BG and BK, which are polynomial algebras. Around 1970 this result was extended to other coefficients through work of Baum, Gugenheim-May, Husemoller-Moore-Stasheff, Munkholm and Wolf. However, the isomorphism was only shown to be additive in this case. I will present a recent result saying that the isomorphism is in fact multiplicative and natural in the pair $(G,K)$, provided that 2 is invertible in the coefficient ring. I will also discuss the main new ideas of the proof, which involve homotopy Gerstenhaber algebras.