  In this paper we deal with the global properties of Willmore surfaces in
spheres via the harmonic conformal Gauss map using loop groups.
  We first derive a global description of those harmonic maps which can be
realized as conformal Gauss maps of some Willmore surfaces (Theorem 3.4,
Theorem 3.11 and Theorem 3.18).
  Then we introduce the DPW procedure for these harmonic maps, and state
appropriate versions of the Iwasawa decomposition and the Birkhoff
decomposition Theorems. In particular, we show how the harmonic maps associated
with Willmore surfaces can be constructed in terms of loop groups.
  The third main result, which has many implications for the case of Willmore
surfaces in spheres, shows that every harmonic map into some non-compact inner
symmetric space $G/K$ induces a harmonic map into the compact dual inner
symmetric space $U/{(U \cap K^\mathbb{C})}$. From this correspondence we obtain
additional information about the global properties of harmonic maps into
non-compact inner symmetric spaces.
  As an illustration of the theory developed in this paper we list examples
(some of which were worked out in separate papers by following the theory of
the present paper). In particular, we present an explicit, unbranched
(isotropic) Willmore sphere in $S^6$ which is not S-Willmore, and thus does not
have a dual Willmore surface. This example gives a negative answer to a long
open problem (originally posed by Ejiri).
