  In this paper, we study the global well-posedness and scattering problem in
the energy space for both focusing and defocusing the Klein-Gordon-Hartree
equation in the spatial dimension $d \geq 3$. The main difficulties are the
absence of an interaction Morawetz-type estimate and of a Lorentz invariance
which enable one to control the momentum. To compensate, we utilize the
strategy derived from concentration compactness ideas, which was first
introduced by Kenig and Merle \cite{KM} to the scattering problem. Furthermore,
employing technique from \cite{Pa2}, we consider a virial-type identity in the
direction orthogonal to the momentum vector so as to control the momentum in
the defocusing case. While in the focusing case, we show that the scattering
holds when the initial data $(u_0,u_1)$ is radial, and the energy
$E(u_0,u_1)<E(\mathcal{W},0)$ and $\|\nabla u_0\|_2^2+\|u_0\|_2^2<\|\nabla
\mathcal{W}\|_2^2+\|\mathcal{W}\|_2^2$, where $\mathcal{W}$ is the ground
state.
