  In Part I, we develop the notions of a Moebius structure and a conformal
Cartan geometry, establish an equivalence between them; we use them in Part II
to study submanifolds of conformal manifolds in arbitrary dimension and
codimension. We obtain Gauss-Codazzi-Ricci equations and a conformal Bonnet
theorem characterizing immersed submanifolds of the conformal n-sphere. These
methods are applied in Part III to study constrained Willmore surfaces,
isothermic surfaces, Guichard surfaces and conformally-flat submanifolds with
flat normal bundle, and their spectral deformations, in arbitrary codimension.
The high point of these applications is a unified theory of Moebius-flat
submanifolds, which include Guichard surfaces and conformally flat
hypersurfaces.
