TITLE: Applications of topological and diferentiable stacks
QUESTION [22 upvotes]: What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well aware of several statements made more beautiful in the language of stacks, but, I'm looking for a concrete application.

REPLY [4 votes]: I should update with a mention of some of my own results in http://arxiv.org/abs/1504.02394: 

There is a proof of Segal's theorem that the classifying space $B\Gamma^q$ of Haefliger's foliation groupoid is homotopy equivalent to classifying space of the discrete monoid of embeddings of $\mathbb{R}^n$ into itself $B\mathbf{Emb}\left(\mathbb{R}^n\right)$ using differentiable stacks (and higher topos theory). (This is theorem 3.7)
You can also use the same machinery to prove the following theorem (Theorem 4.1):

Let $G$ be a Lie group acting almost freely on a manifold $M$. Then the homotopy type of the Borel construction $M\times_G EG$ is the same as the the classifying space of a certain discrete category, whose objects are smooth tranversals, i.e. maps $f:\mathbb{R}^n \to M,$ with $n=\dim M - \dim G$ which are transverse to the $G$-orbits.