I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far:

1. No-wandering-domain Theorem (Sullivan)
2. Theorems of Thurston (Classification of homeomorphism of surfaces, topological characterization of rational maps, hyperbolization theorems for special 3 manifolds)
3. Computer graphics. (Using the various metrics on the Teichmüller spaces as a substitute for Gromov-Hausdorff metric.)
4. String Theory (as elementary particles are modeled by loops, they generate a Riemann surface as they move through time.)
5. Some applications to biology.(Brain morphometry)

Note: I am very sure that this is only a small fraction of what is out there, and I plan to continue to update this list.