  We develop a bubble tree construction and prove compactness results for
$W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with
varying conformal structures whose limit may degenerate, in a compact
Riemannian manifold with uniformly bounded areas and Willmore energies. The
compactness property is applied to construct Willmore type surfaces in compact
Riemannian manifolds. This includes (a) existence of a Willmore 2-sphere in
${\mathbb S}^n$ with at least 2 nonremovable singular points (b) existence of
minimizers of the Willmore functional with prescribed area in a compact
manifold $N$ provided (i) the area is small when genus is 0 and (ii) the area
is close to that of the area minimizing surface of Schoen-Yau and
Sacks-Uhlenbeck in the homotopy class of an incompressible map from a surface
of positive genus to $N$ and $\pi_2(N)$ is trivial (c) existence of smooth
minimizers of the Willmore functional if a Douglas type condition is satisfied.
