TITLE: Important results that use infinite-dimensional manifolds?
QUESTION [29 upvotes]: Are Banach manifolds (or other types of infinite-dimensional manifolds) just curiosities, or have they been utilized to prove some interesting/important results? Where do they turn up? Important examples?

REPLY [5 votes]: David Ebin and Jerry Marsden proved in a 1970 paper in the Annals that the Euler and Navier-Stokes equations are well-posed for short amounts of time.  This result uses the geometry of infinite-dimensional manifolds in a very fundamental way: the configuration space of an incompressible fluid in a container $M$ is the group of volume-preserving diffeomorphisms $\mathrm{Diff}_{\mathrm{vol}}(M)$ of $M$ and this is of course an infinite-dimensional manifold.  The proof then relies on the fact that the Euler equations define a Hamiltonian vector field on this space, and by carefully unwinding the definitions and putting everything in the right Sobolev space you can essentially use existence and uniqueness theorems for ODEs to prove the desired result.
I'm of course glossing over many details.  One of the problems, for instance, when trying to make this idea rigorous is that the convective term $\mathbf{u} \cdot \nabla \mathbf{u}$ in the Euler equations leads to derivative loss, but this can be surmounted by rewriting the equations in Lagrangian form.  If you want to do all this going back and forth in a proper way, you need a good understanding of the underlying geometry of $\mathrm{Diff}_{\mathrm{vol}}(M)$.
In my opinion, this is quite an amazing result, since it is both powerful and conceptually very clear.  There are essentially no hard estimates in the paper, just applications of the Sobolev embedding theorems, Hodge decompositions, properties of vector fields, ....  Global analysis at its best!