I will describe the Teichmüller space of flat metrics on a compact manifold, and the boundary of this space, which consists of flat orbifolds to which the manifold may collapse. The Teichmüller space is described in terms of the isotypic components of the holonomy representation of the underlying manifold. I will prove that every compact flat orbifold can be obtained by collapsing flat manifolds. An application to the Yamabe problem on noncompact manifolds will also be discussed. This is joint work with R. Bettiol (UPenn) and A. Derdzinski (OSU).