In this talk, we discuss the phase diagram of the Hartree-Fock (HF) Homogeneous Electron Gas (HEG), both theoretically and numerically. We first recall the Overhauser spatial symmetry breaking which occurs for T=0 at all densities and we present a new lower bound on the energy gain due to this symmetry breaking. At high density we can prove that this gain can only be exponentially small. Then we focus on the fluid phase of the HF HEG and discuss spin symmetry breaking. We find two regions where the gas is either ferromagnetic or paramagnetic. We can partly justify the paramagnetic phase theoretically. This is joint work with Mathieu Lewin and Christian Hainzl.