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

Bubble tree of a class of conformal mappings and applications to Willmore functional