The Willmore Functional for surfaces has been introduced for the first time almost one century ago in the framework of conformal geometry (it's one dimensional version already appears in the work of Daniel Bernouilli in the XVIII-th century). Maybe because of it's simplicity and the depth of its mathematical relevance, it has since then shown up in various fields of sciences and technology such as cell biology, non-linear elasticity, general elativity...optical design...etc. Critical points to that functional are called Willmore Surfaces. They satisfy the so called Willmore Equation. This equation , despite the elegance of its formulation, is very inappropriate for dealing with analysis questions such as regularity, compactness...etc. We will present a new formulation of the Willmore Euler-Lagrange equation in terms of conservation laws. We shall then explain how this new formulation, together with the Integrability by compensation theory, permit to solve fundamental analysis questions regarding this functional which were untill now totally open : 1) the limits of Palais-Smale sequences to Willmore Lagrangians is Willmore. 2) Weak Willmore Immersions are analytic. 3) The Moduli space of Willmore Torii below 8\pi is strongly compact modulo the Moebius group action in 3 and 4 dimensions. We will conclude by presenting several open problems related to the analysis of Willmore surfaces.