We analyze interactions at high energies in AdS spaces, at large and small AdS curvatures. We then use the results to analyze deep inelastic scattering at small Bjorken x, using the approximate conformal invariance of QCD at high energies. Hard pomeron exchanges are resummed eikonally, restoring unitarity at large values of the phase shift in the dual AdS geometry. At weak coupling this phase is imaginary, corresponding to a black disk in AdS. In this saturated regime, cross sections exhibit geometric scaling and have a simple universal form, which we test against available experimental data for the proton structure function F2(x,Q2). We predict, in particular, the dependence of the cross section on the scaling variable (Q/Qs)2 in the deeply saturated region, where Qs is the usual saturation scale. We find agreement with current data on F2 in the kinematical region defined by Q2 between 0.5 and 10 GeV2, x less than 10E(-2), with an average 6 per cent accuracy. We conclude by discussing the relation of our approach with the commonly used dipole formalism.