The excess of GeV-scale photons observed by the Fermi telescope in the direction of the Galactic Centre is strikingly similar in both rate and spatial distribution to the expectations for annihilating dark matter. I'll discuss dark matter models that can reconcile the Fermi signal with the lack of signals from direct detection and collider experiments. If the Fermi signal is indeed due to dark matter, then the Milky Way's super-massive black hole should give rise to an additional point-source like contribution to the gamma ray signal. The canonical expectation for this point-like source is incompatible with Fermi measurements of the Galactic Centre, and I'll talk about possible astrophysical and particle consequences of this observation.
