  The analysis of the gamma-ray photons collected by the Fermi Large Area
Telescope reveals, after removal of astrophysical background, the existence of
an excess towards the Galactic center. This excess peaks around few GeV, and
its origin is compatible with the gamma-ray flux originating from Dark Matter
annihilation. In this work we take a closer look on this interpretation; we
investigate which kind of Dark Matter, and which type of interactions with the
Standard Model fields are able to reproduce the observed signal. The structure
of the paper is twofold. In the first part, we follow an effective field theory
approach considering both fermionic and scalar Dark Matter. The computation of
the relic density, the constraint imposed from the null result of direct
searches, and the reliability of the effective field theory description allow
us to single out only two viable dim-6 operators in the case of fermionic Dark
Matter. In the second part, we analyze some concrete models. In particular, we
find that the scalar Higgs portal can provide a simple, concrete and realistic
scenario able to explain the GeV excess under scrutiny.
