We prove a bubble-neck decomposition and an energy quantization result for sequences of Willmore surfaces immersed into R^m (m>2) with uniformly bounded energy and non-degenerating conformal type. We deduce the strong compactness (modulo the action of the Moebius group) of closed Willmore surfaces of a given genus below some energy threshold. This is joint-work with Tristan Rivière (ETH Zürich)