A homotopy Gerstenhaber structure on a differential graded algebra is essentially a family of operations defining a multiplication on its bar construction. After a review of the definition, I will explain that the singular cochain algebra of the classifying space of a torus is formal as a homotopy Gerstenhaber algebra. This generalizes to Davis-Januszkiewicz spaces. These formality results allow to prove the formula for the cup product in the cohomology of smooth toric varieties presented in the previous lecture, and it is also crucial for the description of the cohomology ring of homogeneous spaces to be discussed in the next talk. As another application, we describe the cohomology rings of free and based loop spaces of Davis-Januszkiewicz spaces.