I will recall two classical formality theorems in algebraic geometry. The Bogolomov-Tian-Todorov theorem asserts that the Kodaira Spencer Lie algebra of a Calabi-Yau manifold is formal. As a consequence, the deformation theory of a Calabi-Yau manifold is unobstructed. The Deligne-Griffiths-Morgan-Sullivan theorem asserts that the Dolbeault or de Rham algebra of a compact Kähler manifold is formal. As a consequence the real homotopy type of such a manifold is determined by the cohomology with real coefficients. In joint work with Joana Cirici we propose a common generalization of these two theorems. Using work of Baranikov and Kontsevich one can equip a variant of the Dolbeault complex with an action of the operad of compactified moduli spaces of curves of genus zero. This structure determines both the Kodaira-Spencer Lie algebra and the commutative structure of the Dolbeault complex. We prove that this structure is formal in a suitable sense.