  We begin the process of unitarizing the Pomeron at large 't Hooft coupling.
We do so first in the conformal regime, which applies to good accuracy to a
number of real and toy problems in QCD. We rewrite the conformal Pomeron in the
$J$-plane and transverse position space, and then work out the eikonal
approximation to multiple Pomeron exchange. This is done in the context of a
more general treatment of the complex $J$-plane and the geometric consequences
of conformal invariance. The methods required are direct generalizations of our
previous work on single Pomeron exchange and on multiple graviton exchange in
AdS space, and should form a starting point for other investigations. We
consider unitarity and saturation in the conformal regime, noting elastic and
absorptive effects, and exploring where different processes dominate. Our
methods extend to confining theories and we briefly consider the Pomeron kernel
in this context. Though there is important model dependence that requires
detailed consideration, the eikonal approximation indicates that the Froissart
bound is generically both satisfied and saturated.
