  This work explores the celebrated AdS/CFT correspondence in the regime of
high energy scattering in Anti--de Sitter (AdS) spacetime. In particular, we
develop the eikonal approximation to high energy scattering in AdS and explore
its consequences for the dual Conformal Field Theory (CFT).
  Using position space Feynman rules, we rederive the eikonal approximation for
high energy scattering in flat space. Following this intuitive position space
perspective, we then generalize the eikonal approximation for high energy
scattering in AdS and other spacetimes. Remarkably, we are able to resum, in
terms of a generalized phase shift, ladder and cross ladder Witten diagrams
associated to the exchange of an AdS spin j field, to all orders in the
coupling constant.
  By the AdS/CFT correspondence, the eikonal amplitude in AdS is related to the
four point function of CFT primary operators in the regime of large 't Hooft
coupling, including all terms of the 1/N expansion. We then show that the
eikonal amplitude determines the behavior of the CFT four point function for
small values of the cross ratios in a Lorentzian regime and that this controls
its high spin and dimension conformal partial wave decomposition. These results
allow us to determine the anomalous dimension of high spin and dimension double
trace primary operators, by relating it to the AdS eikonal phase shift. Finally
we find that, at large energies and large impact parameters in AdS, the
gravitational interaction dominates all other interactions, as in flat space.
Therefore, the anomalous dimension of double trace operators, associated to
graviton exchange in AdS, yields a universal prediction for CFT's with AdS
gravitational duals.
