The little \(N\)-disks operad, \(\mathcal B\), along with its variants, is an important tool in homotopy theory. It is defined in terms of configurations of disjoint \(N\)-dimensional disks inside the standard unit disk in \(\mathbb{R}^N\) and it was initially conceived for detecting and understanding \(N\)-fold loop spaces. Its many uses now stretch across a variety of disciplines including topology, algebra, and mathematical physics.

In this paper, the authors develop the details of Kontsevich's proof of the formality of little \(N\)-disks operad over the field of real numbers. More precisely, one can consider the singular chains \(\operatorname{C}_*(\mathcal B; \mathbb{R})\) on \(\mathcal B\) as well as the singular homology \(\operatorname{H}_*(\mathcal B; \mathbb{R})\) of \(\mathcal B\). These two objects are operads in the category of chain complexes. The formality then states that there is a zig-zag of quasi-isomorphisms connecting these two operads. The formality also in some sense holds in the category of commutative differential graded algebras. The authors additionally prove a relative version of the formality for the inclusion of the little \(m\)-disks operad in the little \(N\)-disks operad when \(N\geq2m+1\).