We establish that $E_n$-operads satisfy a rational intrinsic formality theorem for $n\geq 3$. We gain our results in the category of Hopf cooperads in cochain graded dg-modules which defines a model for the rational homotopy of operads in spaces. We consider, in this context, the dual cooperad of the $n$-Poisson operad $\mathsf{Pois}_n^c$, which represents the cohomology of the operad of little $n$-discs $\mathsf{D}_n$. We assume $n\geq 3$. We explicitly prove that a Hopf cooperad in cochain graded dg-modules $\mathsf{K}$ is weakly-equivalent (quasi-isomorphic) to $\mathsf{Pois}_n^c$ as a Hopf cooperad as soon as we have an isomorphism at the cohomology level $H^*(\mathsf{K})\simeq\mathsf{Pois}_n^c$ when $4\nmid n$. We just need the extra assumption that $\mathsf{K}$ is equipped with an involutive isomorphism mimicking the action of a hyperplane reflection on the little $n$-discs operad in order to extend this formality statement in the case $4\mid n$. We deduce from these results that any operad in simplicial sets $\mathsf{P}$ which satisfies the relation $H^*(\mathsf{P},\mathbb{Q})\simeq\mathsf{Pois}_n^c$ in rational cohomology (and an analogue of our extra involution requirement in the case $4\mid n$) is rationally weakly equivalent to an operad in simplicial sets $LG_{\bullet}(\mathsf{Pois}_n^c)$ which we determine from the $n$-Poisson cooperad $\mathsf{Pois}_n^c$. We also prove that the morphisms $\iota: \mathsf{D}_m\rightarrow\mathsf{D}_n$, which link the little discs operads together, are rationally formal as soon as $n-m\geq 2$. These results enable us to retrieve the (real) formality theorems of Kontsevich by a new approach, and to sort out the question of the existence of formality quasi-isomorphisms defined over the rationals (and not only over the reals) in the case of the little discs operads of dimension $n\geq 3$.

We express the rational homotopy type of the mapping spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ of the little discs operads in terms of graph complexes. Using known facts about the graph homology this allows us to compute the rational homotopy groups in low degrees, and construct infinite series of non-trivial homotopy classes in higher degrees. Furthermore we show that for $n-m>2$, the spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ and $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n)$ are simply connected and rationally equivalent. As application we determine the rational homotopy type of the deloopings of spaces of long embeddings. Some of the results hold also for mapping spaces $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$, $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n)$, $n-m\geq 2$, of the truncated little discs operads, which allows one to determine rationally the delooping of the Goodwillie-Weiss tower for the spaces of long embeddings.