The ADM mass is a metric scalar quantity computed at infinity for an Asymptotically Euclidean manifold. It carries a physical meaning as it isolates the mass of a black hole, and more remarkably a geometric importance (with positivity and rigidity results) as well as an analytical role (it intervenes in the development of the Green function of the Yamabe operator). We will in this talk introduce this ADM mass using a conserved quantity approach to general relativity and explain why it can be expected to play such a pivotal role.

We will then apply the same method to introduce a mass linked to another gravitational theory, and then study the geometric and analytical importance of this new mass. In particular, we will link it to a fundamental higher order geometric quantity: the Q-curvature.