One big recent example would be Lafforgue's proof of the Langlands correspondence for $GL_n$ of function fields (<http://arxiv.org/abs/math.NT/0212399>), which uses stacks of schtukas. It is similar to Drinfel'd's proof for $GL_2$, but with the moduli space being an essential component.  

More readable versions, with additional context, are given by Lafforgue's advisor Gerard Laumon (<http://arxiv.org/abs/math.AG/0003131> if you can read French) and by his student Ngo Dac Tuan (MR2402699 on MathSciNet, or <http://www.impan.pl/~pragacz/download/Ngo.pdf>)