  In the search for appropriate discretizations of surface theory it is crucial
to preserve such fundamental properties of surfaces as their invariance with
respect to transformation groups. We discuss discretizations based on M\"obius
invariant building blocks such as circles and spheres. Concrete problems
considered in these lectures include the Willmore energy as well as conformal
and curvature line parametrizations of surfaces. In particular we discuss
geometric properties of a recently found discrete Willmore energy. The
convergence to the smooth Willmore functional is shown for special refinements
of triangulations originating from a curvature line parametrization of a
surface. Further we treat special classes of discrete surfaces such as
isothermic and minimal. The construction of these surfaces is based on the
theory of circle patterns, in particular on their variational description.
