  We generalize Regge theory to correlation functions in conformal field
theories. This is done by exploring the analogy between Mellin amplitudes in
AdS/CFT and S-matrix elements. In the process, we develop the conformal partial
wave expansion in Mellin space, elucidating the analytic structure of the
partial amplitudes. We apply the new formalism to the case of four point
correlation functions between protected scalar operators in N=4 Super Yang
Mills, in cases where the Regge limit is controlled by the leading twist
operators associated to the pomeron-graviton Regge trajectory. At weak
coupling, we are able to predict to arbitrary high order in the 't Hooft
coupling the behaviour near J=1 of the OPE coefficients C_{OOJ} between the
external scalars and the spin J leading twist operators. At strong coupling, we
use recent results for the anomalous dimension of the leading twist operators
to improve current knowledge of the AdS graviton Regge trajectory - in
particular, determining the next and next to next leading order corrections to
the intercept. Finally, by taking the flat space limit and considering the
Virasoro-Shapiro S-matrix element, we compute the strong coupling limit of the
OPE coefficient C_{LLJ} between two Lagrangians and the leading twist operators
of spin J.
